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

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">C</font> is  <a href="cat_4.html#V4" title="CAT_4:attr.4">with_finite_coproduct</a>  iff (  ex <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <font color="Olive" title="b1">a</font> is  <a href="cat_1.html#V7" title="CAT_1:attr.7">initial</a>  &amp; (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">i1</font>, <font color="Olive" title="b5">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b4">i1</font>,<font color="Olive" title="b5">i2</font> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:62"/>
( <font color="Maroon" title="c1">C</font> is  <a href="cat_4.html#V4" title="CAT_4:attr.4">with_finite_coproduct</a>  implies (  ex <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <font color="Olive" title="b1">a</font> is  <a href="cat_1.html#V7" title="CAT_1:attr.7">initial</a>  &amp; (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">i1</font>, <font color="Olive" title="b5">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b4">i1</font>,<font color="Olive" title="b5">i2</font> ) ) ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <font color="Olive" title="b1">a</font> is  <a href="cat_1.html#V7" title="CAT_1:attr.7">initial</a>  &amp; (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">i1</font>, <font color="Olive" title="b5">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b4">i1</font>,<font color="Olive" title="b5">i2</font> ) ) implies <font color="Maroon" title="c1">C</font> is  <a href="cat_4.html#V4" title="CAT_4:attr.4">with_finite_coproduct</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c2">F</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></span>;<br/>
<a NAME="E2:62_1"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:62_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
 for <font color="Olive" title="b1">I</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <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="Olive" title="b1">I</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">I</font> st <br/>(  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b4">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b4">F9</font> )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_4.html#D20" target="_self" title="CAT_4:def.20">CAT_4:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <font color="Olive" title="b1">a</font> is  <a href="cat_1.html#V7" title="CAT_1:attr.7">initial</a>  &amp; (  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">i1</font>, <font color="Olive" title="b5">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b4">i1</font>,<font color="Olive" title="b5">i2</font> ) ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c4">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c3">a</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <span class="kw"> such that </span><br/><a NAME="E5:62_1"/>
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c4">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">and </span><br/><a NAME="E6:62_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c4">F9</font>
 ;<br/>
<a NAME="E7:62_1"/>
<font color="Maroon" title="c3">a</font> is  <a href="cat_1.html#V7" title="CAT_1:attr.7">initial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="cat_3.html#T75" title="CAT_3:th.75">CAT_3:75</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:62_1"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <font color="Olive" title="b1">a</font> is  <a href="cat_1.html#V7" title="CAT_1:attr.7">initial</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">i1</font>, <font color="Olive" title="b5">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b4">i1</font>,<font color="Olive" title="b5">i2</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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="c1">C</font> ex <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b1">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">F</font> = (<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c9">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c8">c</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span><span class="kw"> such that </span><br/><a NAME="E10:62_1"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c9">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> (<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)
 <span class="kw">and </span><br/><a NAME="E11:62_1"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c9">F9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c8">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c10">i1</font> = <font color="Maroon" title="c9">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c8">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c10">i1</font>,<font color="Olive" title="b1">i2</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c11">i2</font> = <font color="Maroon" title="c9">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c8">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">i2</font> )</span><br/>

<a NAME="E12:62_1"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E13:62_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>;<br/>
<a NAME="E14:62_1"/>
( <span class="p1">(<span class="default">(<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <span class="p1">(<span class="default">(<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T3" title="CAT_3:th.3">CAT_3:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:62_1"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:62_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E2:62_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E12:62_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp; <font color="Maroon" title="c8">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">i2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E16:62_1"/>
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E12:62_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="cat_3.html#T62" title="CAT_3:th.62">CAT_3:62</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">i2</font></span><br/>

<a NAME="E17:62_1"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">F9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>;<br/>
<a NAME="E18:62_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> (<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>,1) <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> (<font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">i2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:62_1"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:62_1"/>
<font color="Maroon" title="c8">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:62_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="cat_3.html#T79" title="CAT_3:th.79">CAT_3:79</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c2">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><a NAME="E3:62"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c2">a</font> is  <a href="cat_1.html#V7" title="CAT_1:attr.7">initial</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <br/> for <font color="Olive" title="b3">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b4">i1</font>, <font color="Olive" title="b5">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> or  not  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> or  not  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b3">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b4">i1</font>,<font color="Olive" title="b5">i2</font> ) or <font color="Maroon" title="c1">C</font> is  <a href="cat_4.html#V4" title="CAT_4:attr.4">with_finite_coproduct</a>  )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">I</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <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="Olive" title="b1">I</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> ex <font color="Olive" title="b3">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">I</font> st <br/>(  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b4">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b4">F9</font> );<br/>
<span class="kw">assume </span><a NAME="E4:62"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">i1</font>, <font color="Olive" title="b5">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b4">i1</font>,<font color="Olive" title="b5">i2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> is  <a href="cat_4.html#V4" title="CAT_4:attr.4">with_finite_coproduct</a> </span><br/>

<a NAME="E5:62"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:62_2"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">I</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">I</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> ex <font color="Olive" title="b2">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b3">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive" title="b2">a</font>,<font color="Maroon" title="c4">I</font> st <br/>(  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b3">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b3">F9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">I</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  ex <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">I</font> st <br/>(  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b2">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b2">F9</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E2:62_2"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">I</font> st <br/>(  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b2">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b2">F9</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">Ix</font> = <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">sx</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">G</font> = <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></span>;<br/>
<a NAME="E6:62_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<span class="kw">then </span> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">Ix</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">sx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" title="CARD_1:th.30">CARD_1:30</a></span>
<br/>.= 
<font color="Maroon" title="c3">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="xcmplx_1.html#T26" title="XCMPLX_1:th.26">XCMPLX_1:26</a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c11">G9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span><span class="kw"> such that </span><br/><a NAME="E9:62_2"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c11">G9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">G</font>
 <span class="kw">and </span><br/><a NAME="E10:62_2"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c11">G9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2"><span class="lab"><font color="Green" title="E32">A11</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">b</font> = <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c14">i1</font>, <font color="Maroon" title="c15">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E12:62_2"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">and </span><br/><a NAME="E13:62_2"/><span class="lab"><font color="Green" title="E37">A16</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c15">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>
 <span class="kw">and </span><br/><a NAME="E14:62_2"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c</font>
 <span class="kw">and </span><br/><a NAME="E15:62_2"/><span class="lab"><font color="Green" title="E39">A18</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c15">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c</font>
 <span class="kw">and </span><br/><a NAME="E16:62_2"/><span class="lab"><font color="Green" title="E40">A19</font></span>: 
<font color="Maroon" title="c13">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c14">i1</font>,<font color="Maroon" title="c15">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E30">A9</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">F9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c15">i2</font></span>)</span>;<br/>
<a NAME="E17:62_2"/><span class="lab"><font color="Green" title="E41">A20</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c15">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E18:62_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c15">i2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c15">i2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T17" title="FUNCT_4:th.17">FUNCT_4:17</a></span>;<br/>
<a NAME="E19:62_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A21</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c15">i2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E20:62_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E43">A22</font></span>: <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c15">i2</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c4">I</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>
<br/>.= 
<font color="Maroon" title="c4">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="zfmisc_1.html#T40" title="ZFMISC_1:th.40">ZFMISC_1:40</a></span>
;<br/>
<a NAME="E22:62_2"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c15">i2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">F9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c15">i2</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">I</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:62_2"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<a NAME="E24:62_2"/><span class="lab"><font color="Green" title="E44">A23</font></span>: 
 <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c11">G9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c10">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D15" title="CAT_3:def.15">CAT_3:def 15</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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="c17">F9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:62_2_3"/><span class="lab"><font color="Green" title="E45">A24</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:62_2_3_1_1"/>
( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_2_3_1_1_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</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="c17">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font></span><br/><div class="add"><a NAME="E2:62_2_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A25</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> = 
<font color="Maroon" title="c17">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_3"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c15">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E2:62_2_3_1_1_1"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c15">i2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_2_3_1_1_1"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> = 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:62_2"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E1:62_2_3"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_3"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="cat_3.html#D2" title="CAT_3:def.2">CAT_3:def 2</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_2_3_1_1_2"/><span class="lab"><font color="Green" title="E46">A26</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font></span><br/><div class="add"><a NAME="E2:62_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A27</font></span>: 
 not <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:62_2_3_1_1_2"/><span class="lab"><font color="Green" title="E48">A28</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_3"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E1:62_2_3_1_1_2"><span class="lab"><font color="Green" title="E46">A26</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></span>;<br/><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> = 
<font color="Maroon" title="c17">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_3"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E21:62_2"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="txt" href="#E1:62_2_3"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E2:62_2_3_1_1_2"><span class="lab"><font color="Green" title="E47">A27</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_2_3_1_1_2"><span class="lab"><font color="Green" title="E48">A28</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> = 
 <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="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_3"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="cat_3.html#D2" title="CAT_3:def.2">CAT_3:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_2_3_1_1_2"><span class="lab"><font color="Green" title="E48">A28</font></span></a>, <a class="ref" href="cat_3.html#D2" title="CAT_3:def.2">CAT_3:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E14:62_2"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="txt" href="#E24:62_2"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="ref" href="cat_3.html#T17" title="CAT_3:th.17">CAT_3:17</a></span>
<br/>.= 
<font color="Maroon" title="c13">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_2_3_1_1_2"><span class="lab"><font color="Green" title="E48">A28</font></span></a>, <a class="ref" href="cat_3.html#T2" title="CAT_3:th.2">CAT_3:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_3"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:62_2_3"/>
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_3"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E26:62_2"/><span class="kw">then </span>
 <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">F9</font> = <font color="Maroon" title="c17">F9</font> as    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c13">c</font>,<font color="Maroon" title="c4">I</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D15" title="CAT_3:def.15">CAT_3:def 15</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c13">c</font>,<font color="Maroon" title="c4">I</font> st <br/>(  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b1">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c13">c</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b1">F9</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c18">F9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c13">c</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c18">F9</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> implies <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:62_2_4"/><span class="lab"><font color="Green" title="E45">A29</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:62_2_4_1_1"/>
( <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> or <font color="Maroon" title="c19">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_2_4_1_1_1"/><span class="lab"><font color="Green" title="E46">A30</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font></span><br/><div class="add"><a NAME="E2:62_2_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A31</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font> = 
<font color="Maroon" title="c18">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_4"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c15">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E2:62_2_4_1_1_1"><span class="lab"><font color="Green" title="E47">A31</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c15">i2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_2_4_1_1_1"><span class="lab"><font color="Green" title="E47">A31</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:62_2_4_1_1_1"/>
<span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:62_2"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="txt" href="#E1:62_2_4"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="txt" href="#E1:62_2_4_1_1_1"><span class="lab"><font color="Green" title="E46">A30</font></span></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_2_4_1_1_2"/><span class="lab"><font color="Green" title="E46">A32</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font></span><br/><div class="add"><a NAME="E2:62_2_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A33</font></span>: 
 not <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:62_2_4_1_1_2"/><span class="lab"><font color="Green" title="E48">A34</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_4"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="txt" href="#E1:62_2_4_1_1_2"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></span>;<br/><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font> = 
<font color="Maroon" title="c18">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_4"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E21:62_2"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="txt" href="#E1:62_2_4"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="txt" href="#E2:62_2_4_1_1_2"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_2_4_1_1_2"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</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="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_4"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_2_4_1_1_2"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c9">G</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:62_2"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E12:62_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E24:62_2"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="ref" href="cat_3.html#T17" title="CAT_3:th.17">CAT_3:17</a></span>
<br/>.= 
<font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_2_4_1_1_2"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_2_4_1_1_2"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_4"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:62_2_4"/>
<span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E29:62_2"/><span class="lab"><font color="Green" title="E45">A35</font></span>: 
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T1" title="CAT_3:th.1">CAT_3:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">c</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c18">F9</font></span><br/>

<span class="kw">thus </span><a NAME="E30:62_2"/>
<font color="Maroon" title="c18">F9</font> is    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c13">c</font>,<font color="Maroon" title="c4">I</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_3.html#D16" title="CAT_3:def.16">CAT_3:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive">b<sub>2</sub></font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c4">I</font>  holds <br/>(  not  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font> <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">b<sub>2</sub></font> or  ex <font color="Olive">b<sub>3</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive">b<sub>3</sub></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="c13">c</font>,<font color="Olive">b<sub>1</sub></font>) &amp; (  for <font color="Olive">b<sub>4</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not <font color="Olive">b<sub>4</sub></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="c13">c</font>,<font color="Olive">b<sub>1</sub></font>) or ( (  ex <font color="Olive">b<sub>5</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  not <font color="Olive">b<sub>4</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>5</sub></font> ) or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> ) &amp; (  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> or  for <font color="Olive">b<sub>5</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <font color="Olive">b<sub>4</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>5</sub></font> ) ) ) ) ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c19">d</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c19">d</font>,<font color="Maroon" title="c4">I</font>  holds <br/>(  not  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font> <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">b<sub>1</sub></font> or  ex <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive">b<sub>2</sub></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="c13">c</font>,<font color="Maroon" title="c19">d</font>) &amp; (  for <font color="Olive">b<sub>3</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not <font color="Olive">b<sub>3</sub></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="c13">c</font>,<font color="Maroon" title="c19">d</font>) or ( (  ex <font color="Olive">b<sub>4</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  not <font color="Olive">b<sub>3</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>4</sub></font> ) or <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> ) &amp; (  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> or  for <font color="Olive">b<sub>4</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <font color="Olive">b<sub>3</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>4</sub></font> ) ) ) ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c20">F99</font> be    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c19">d</font>,<font color="Maroon" title="c4">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font> <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="c20">F99</font> or  ex <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive">b<sub>1</sub></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="c13">c</font>,<font color="Maroon" title="c19">d</font>) &amp; (  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></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="c13">c</font>,<font color="Maroon" title="c19">d</font>) or ( (  ex <font color="Olive">b<sub>3</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  not <font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font> ) or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; (  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> or  for <font color="Olive">b<sub>3</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font> ) ) ) ) ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E31:62_2"/><span class="lab"><font color="Green" title="E46">A36</font></span>: 
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font> <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="c20">F99</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive">b<sub>1</sub></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="c13">c</font>,<font color="Maroon" title="c19">d</font>) &amp; (  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></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="c13">c</font>,<font color="Maroon" title="c19">d</font>) or ( (  ex <font color="Olive">b<sub>3</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  not <font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font> ) or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; (  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> or  for <font color="Olive">b<sub>3</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font> ) ) ) ) ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c21">G99</font> = <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c21">G99</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c19">d</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:62_2_5"/><span class="lab"><font color="Green" title="E47">A37</font></span>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c21">G99</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c19">d</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font></span><br/><span class="kw">then </span><font color="Maroon" title="c21">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font> = 
<font color="Maroon" title="c21">G99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c20">F99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_5"><span class="lab"><font color="Green" title="E47">A37</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_5"><span class="lab"><font color="Green" title="E47">A37</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c21">G99</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font> = 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_5"><span class="lab"><font color="Green" title="E47">A37</font></span></a>, <a class="ref" href="cat_3.html#D2" title="CAT_3:def.2">CAT_3:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c19">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_5"><span class="lab"><font color="Green" title="E47">A37</font></span></a>, <a class="ref" href="cat_3.html#T62" title="CAT_3:th.62">CAT_3:62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c19">d</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_5"><span class="lab"><font color="Green" title="E47">A37</font></span></a>, <a class="ref" href="cat_3.html#T2" title="CAT_3:th.2">CAT_3:2</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E34:62_2"/><span class="kw">then </span>
 <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c21">G99</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c19">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T1" title="CAT_3:th.1">CAT_3:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">G99</font> = <font color="Maroon" title="c21">G99</font> as    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c19">d</font>,<font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D15" title="CAT_3:def.15">CAT_3:def 15</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">y</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> <font color="Maroon" title="c22">G99</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:62_2_6"/><span class="lab"><font color="Green" title="E47">A38</font></span>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">y</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> <font color="Maroon" title="c22">G99</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">y</font></span><br/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A39</font></span>: <font color="Maroon" title="c9">G</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">y</font> = 
<font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_6"><span class="lab"><font color="Green" title="E47">A38</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_6"><span class="lab"><font color="Green" title="E47">A38</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
;<br/><font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">y</font> = 
<font color="Maroon" title="c20">F99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_6"><span class="lab"><font color="Green" title="E47">A38</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c22">G99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_6"><span class="lab"><font color="Green" title="E47">A38</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c22">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_6"><span class="lab"><font color="Green" title="E47">A38</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">y</font> = 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:62_2"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E29:62_2"><span class="lab"><font color="Green" title="E45">A35</font></span></a>, <a class="txt" href="#E31:62_2"><span class="lab"><font color="Green" title="E46">A36</font></span></a>, <a class="txt" href="#E1:62_2_6"><span class="lab"><font color="Green" title="E47">A38</font></span></a>, <a class="txt" href="#E2:62_2_6"><span class="lab"><font color="Green" title="E48">A39</font></span></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c22">G99</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_6"><span class="lab"><font color="Green" title="E47">A38</font></span></a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E37:62_2"/><span class="kw">then </span>
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c11">G9</font> <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="c22">G99</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T1" title="CAT_3:th.1">CAT_3:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">h9</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E39:62_2"/><span class="lab"><font color="Green" title="E47">A40</font></span>: 
<font color="Maroon" title="c23">h9</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="c10">a</font>,<font color="Maroon" title="c19">d</font>)
 <span class="kw">and </span><br/><a NAME="E40:62_2"/><span class="lab"><font color="Green" title="E48">A41</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">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="c10">a</font>,<font color="Maroon" title="c19">d</font>) holds <br/>( (  for <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> 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"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">y</font> ) iff <font color="Maroon" title="c23">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:62_2"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="cat_3.html#D16" title="CAT_3:def.16">CAT_3:def 16</a></span>;<br/>
<a NAME="E41:62_2"/><span class="lab"><font color="Green" title="E49">A42</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">g</font> = <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>;<br/>
<a NAME="E42:62_2"/><span class="lab"><font color="Green" title="E50">A43</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="cat_3.html#T62" title="CAT_3:th.62">CAT_3:62</a></span>;<br/>
<span class="lab"><font color="Green" title="E51">A44</font></span>: <font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> = 
<font color="Maroon" title="c18">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c15">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E41:62_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c15">i2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:62_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
;<br/>
<span class="kw">then </span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c15">i2</font> = 
<span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c18">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></span>
<br/>.= 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E31:62_2"><span class="lab"><font color="Green" title="E46">A36</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="cat_3.html#D1" title="CAT_3:def.1">CAT_3:def 1</a></span>
;<br/>
<a NAME="E45:62_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c15">i2</font></span>)</span>,<font color="Maroon" title="c19">d</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:62_2"><span class="lab"><font color="Green" title="E50">A43</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E47:62_2"/><span class="lab"><font color="Green" title="E52">A45</font></span>: 
<font color="Maroon" title="c25">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="c13">c</font>,<font color="Maroon" title="c19">d</font>)
 <span class="kw">and </span><br/><a NAME="E48:62_2"/><span class="lab"><font color="Green" title="E53">A46</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">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="c13">c</font>,<font color="Maroon" title="c19">d</font>) holds <br/>( ( <font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">h9</font> &amp; <font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c15">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> ) iff <font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E16:62_2"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E39:62_2"><span class="lab"><font color="Green" title="E47">A40</font></span></a>, <a class="ref" href="cat_3.html#D17" title="CAT_3:def.17">CAT_3:def 17</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c25">h</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">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="c13">c</font>,<font color="Maroon" title="c19">d</font>) &amp; (  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></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="c13">c</font>,<font color="Maroon" title="c19">d</font>) or ( (  ex <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  not <font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font> ) or <font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; (  not <font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or  for <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E49:62_2"/>
<font color="Maroon" title="c25">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="c13">c</font>,<font color="Maroon" title="c19">d</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:62_2"><span class="lab"><font color="Green" title="E52">A45</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></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="c13">c</font>,<font color="Maroon" title="c19">d</font>) or ( (  ex <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  not <font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font> ) or <font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; (  not <font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or  for <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c26">k</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c26">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="c13">c</font>,<font color="Maroon" title="c19">d</font>) or ( (  ex <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  not <font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font> ) or <font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">k</font> ) &amp; (  not <font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">k</font> or  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E50:62_2"/><span class="lab"><font color="Green" title="E54">A47</font></span>: 
<font color="Maroon" title="c26">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="c13">c</font>,<font color="Maroon" title="c19">d</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  ex <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp;  not <font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font> ) or <font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">k</font> ) &amp; (  not <font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">k</font> or  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E51:62_2"/>
( (  for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> holds <br/><font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font> ) implies <font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">k</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">k</font> or  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_9"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:62_2_9"/><span class="lab"><font color="Green" title="E55">A48</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:62_2"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="ref" href="cat_1.html#T1" title="CAT_1:th.1">CAT_1:1</a></span>;<br/>
<a NAME="E2:62_2_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A49</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c14">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E14:62_2"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="ref" href="cat_1.html#T17" title="CAT_1:th.17">CAT_1:17</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:62_2_9"/><span class="lab"><font color="Green" title="E57">A50</font></span>: 
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> holds <br/><font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">k</font></span><br/>

<a NAME="E4:62_2_9"/><span class="lab"><font color="Green" title="E58">A51</font></span>: 
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c14">i1</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c27">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c14">i1</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:62_2_9_1"/><span class="lab"><font color="Green" title="E59">A52</font></span>: 
<font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c14">i1</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span><br/>

<a NAME="E2:62_2_9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A53</font></span>: 
 not <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="lab"><font color="Green" title="E61">A54</font></span>: <font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font> = 
<font color="Maroon" title="c18">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_9_1"><span class="lab"><font color="Green" title="E59">A52</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E21:62_2"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="txt" href="#E1:62_2_9_1"><span class="lab"><font color="Green" title="E59">A52</font></span></a>, <a class="txt" href="#E2:62_2_9_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_9_1"><span class="lab"><font color="Green" title="E59">A52</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
;<br/>
<a NAME="E4:62_2_9_1"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_9_1"><span class="lab"><font color="Green" title="E59">A52</font></span></a>, <a class="ref" href="cat_3.html#T62" title="CAT_3:th.62">CAT_3:62</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c14">i1</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span> = 
<font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E14:62_2"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="txt" href="#E1:62_2_9"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>
<br/>.= 
<font color="Maroon" title="c26">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="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_9_1"><span class="lab"><font color="Green" title="E59">A52</font></span></a>, <a class="ref" href="cat_3.html#D6" title="CAT_3:def.6">CAT_3:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_2_9"><span class="lab"><font color="Green" title="E57">A50</font></span></a>, <a class="txt" href="#E1:62_2_9_1"><span class="lab"><font color="Green" title="E59">A52</font></span></a>, <a class="txt" href="#E3:62_2_9_1"><span class="lab"><font color="Green" title="E61">A54</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c20">F99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_9_1"><span class="lab"><font color="Green" title="E59">A52</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c22">G99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_9_1"><span class="lab"><font color="Green" title="E59">A52</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c22">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_9_1"><span class="lab"><font color="Green" title="E59">A52</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:62_2_9"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:62_2"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="ref" href="cat_1.html#T1" title="CAT_1:th.1">CAT_1:1</a></span>;<br/>
<a NAME="E6:62_2_9"/><span class="kw">then </span>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c14">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:62_2"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="txt" href="#E1:62_2_9"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="ref" href="cat_1.html#T17" title="CAT_1:th.17">CAT_1:17</a></span>;<br/>
<a NAME="E7:62_2_9"/><span class="kw">then </span>
<font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c14">i1</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="c10">a</font>,<font color="Maroon" title="c19">d</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_2_9"><span class="lab"><font color="Green" title="E56">A49</font></span></a></span>;<br/>
<a NAME="E8:62_2_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A55</font></span>: 
<font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">h9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:62_2"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="txt" href="#E4:62_2_9"><span class="lab"><font color="Green" title="E58">A51</font></span></a></span>;<br/>
<a NAME="E9:62_2_9"/>
<font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c15">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_2"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E43:62_2"><span class="lab"><font color="Green" title="E51">A44</font></span></a>, <a class="txt" href="#E3:62_2_9"><span class="lab"><font color="Green" title="E57">A50</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:62_2_9"/>
<font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:62_2"><span class="lab"><font color="Green" title="E53">A46</font></span></a>, <a class="txt" href="#E50:62_2"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="txt" href="#E8:62_2_9"><span class="lab"><font color="Green" title="E59">A55</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E52:62_2"/><span class="lab"><font color="Green" title="E55">A56</font></span>: 
<font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c27">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E53:62_2"/><span class="lab"><font color="Green" title="E56">A57</font></span>: 
<font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_10"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_10_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:62_2_10_1"/>
( <font color="Maroon" title="c27">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> or <font color="Maroon" title="c27">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_10_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_2_10_1_1"/><span class="lab"><font color="Green" title="E57">A58</font></span>: 
<font color="Maroon" title="c27">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span><br/><div class="add"><a NAME="E2:62_2_10_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A59</font></span>: 
<font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font> = 
<font color="Maroon" title="c18">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:62_2"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c15">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E2:62_2_10_1_1"><span class="lab"><font color="Green" title="E58">A59</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c15">i2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_2_10_1_1"><span class="lab"><font color="Green" title="E58">A59</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:62_2_10_1_1"/>
<font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:62_2"><span class="lab"><font color="Green" title="E53">A46</font></span></a>, <a class="txt" href="#E50:62_2"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="txt" href="#E52:62_2"><span class="lab"><font color="Green" title="E55">A56</font></span></a>, <a class="txt" href="#E1:62_2_10_1_1"><span class="lab"><font color="Green" title="E57">A58</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_10_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_2_10_1_2"/><span class="lab"><font color="Green" title="E57">A60</font></span>: 
<font color="Maroon" title="c27">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span><br/><div class="add"><a NAME="E2:62_2_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A61</font></span>: 
 not <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:62_2_10_1_2"/><span class="lab"><font color="Green" title="E59">A62</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:62_2"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="ref" href="cat_1.html#T1" title="CAT_1:th.1">CAT_1:1</a></span>;<br/><a NAME="E4:62_2_10_1_2"/><span class="lab"><font color="Green" title="E60">A63</font></span>: 
<font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:62_2"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="txt" href="#E1:62_2_10_1_2"><span class="lab"><font color="Green" title="E57">A60</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></span>;<br/><a NAME="E5:62_2_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A64</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T62" title="CAT_3:th.62">CAT_3:62</a></span>;<br/><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font> = 
<font color="Maroon" title="c18">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:62_2"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:62_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E21:62_2"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="txt" href="#E53:62_2"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="txt" href="#E2:62_2_10_1_2"><span class="lab"><font color="Green" title="E58">A61</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_2_10_1_2"><span class="lab"><font color="Green" title="E60">A63</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c14">i1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_2_10_1_2"><span class="lab"><font color="Green" title="E60">A63</font></span></a>, <a class="ref" href="cat_3.html#D6" title="CAT_3:def.6">CAT_3:def 6</a></span>
;<br/><span class="kw">hence </span><font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c14">i1</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E14:62_2"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="txt" href="#E3:62_2_10_1_2"><span class="lab"><font color="Green" title="E59">A62</font></span></a>, <a class="txt" href="#E5:62_2_10_1_2"><span class="lab"><font color="Green" title="E61">A64</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>
<br/>.= 
<font color="Maroon" title="c23">h9</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:62_2"><span class="lab"><font color="Green" title="E53">A46</font></span></a>, <a class="txt" href="#E50:62_2"><span class="lab"><font color="Green" title="E54">A47</font></span></a>, <a class="txt" href="#E52:62_2"><span class="lab"><font color="Green" title="E55">A56</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c22">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:62_2"><span class="lab"><font color="Green" title="E47">A40</font></span></a>, <a class="txt" href="#E40:62_2"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="txt" href="#E4:62_2_10_1_2"><span class="lab"><font color="Green" title="E60">A63</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c22">G99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_2_10_1_2"><span class="lab"><font color="Green" title="E60">A63</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c20">F99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_2_10_1_2"><span class="lab"><font color="Green" title="E60">A63</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:62_2"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="ref" href="funct_2.html#D13" title="FUNCT_2:def.13">FUNCT_2:def 13</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E55:62_2"/>
<font color="Maroon" title="c26">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">let </span><font color="Maroon" title="c3">I</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_4.html#D20" target="_self" title="CAT_4:def.20">CAT_4:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">I</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b2">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b3">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive" title="b2">a</font>,<font color="Maroon" title="c3">I</font> st <br/>(  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b3">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b3">F9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">I</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font> st <br/>(  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b2">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b2">F9</font> )</span><br/>



<a NAME="E6:62"/><span class="lab"><font color="Green" title="E32">A65</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">I</font>
 
;<br/>
<a NAME="E7:62"/><span class="lab"><font color="Green" title="E33">A66</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">I</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">I</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  holds <br/> ex <font color="Olive" title="b2">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b3">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive" title="b2">a</font>,<font color="Maroon" title="c5">I</font> st <br/>(  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b3">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b3">F9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">I</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  ex <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c5">I</font> st <br/>(  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b2">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b2">F9</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:62_3"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c5">I</font> st <br/>(  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b2">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b2">F9</font> )</span><br/>

<a NAME="E2:62_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A67</font></span>: 
<font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E3:62_3"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">F9</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">I</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_3"><span class="lab"><font color="Green" title="E34">A67</font></span></a>, <a class="ref" href="cat_3.html#T63" title="CAT_3:th.63">CAT_3:63</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">a</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">I</font> st <br/>(  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b1">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c2">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b1">F9</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">F9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c7">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c2">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c7">F9</font> )</span><br/>

<a NAME="E5:62_3"/>
 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="c5">I</font> holds <br/><span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c7">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</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>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:62_3"/>
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c7">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T1" title="CAT_3:th.1">CAT_3:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c7">F9</font></span><br/>

<span class="kw">thus </span><a NAME="E7:62_3"/>
<font color="Maroon" title="c2">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c7">F9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E2:62_3"><span class="lab"><font color="Green" title="E34">A67</font></span></a>, <a class="ref" href="cat_3.html#T75" title="CAT_3:th.75">CAT_3:75</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:62"/>
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E33">A66</font></span></a>, <a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E31">A10</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:62"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font> st <br/>(  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b2">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b2">F9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E32">A65</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
