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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>thus </b><a NAME="E1:89"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="cat_3.html#R1" target="_self">is_a_product_wrt</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a>  <a href="cat_3.html#R3" target="_self">is_a_coproduct_wrt</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:89_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="cat_3.html#R1" target="_self">is_a_product_wrt</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:89_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="cat_3.html#D15" target="_self">Def15</a></i>;<br/>
<b>hence </b><a NAME="E3:89_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a>  is    <a href="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> ,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="cat_3.html#T74" target="_self">Th74</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D18" target="_self">CAT_3:def 18</a><br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E4:89_1"/><i><font color="Green">E52</font></i>: 
 <a href="cat_3.html#K3" target="_self">doms</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="cat_3.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>6</sub></font> as    <a href="cat_3.html#M1" target="_self">Projections_family</a> of  <a href="oppcat_1.html#K4">opp</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="cat_3.html#T75" target="_self">Th75</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:89_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> = 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cat_3.html#D4" target="_self">Def4</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T9">OPPCAT_1:9</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E1:89_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="cat_3.html#E4:89_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="cat_3.html#E1:89_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cat_3.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E1:89_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cat_3.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T10">OPPCAT_1:10</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E1:89_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cat_3.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="cat_3.html#E1:89_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cat_3.html#D4" target="_self">Def4</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E7:89_1"/><b>then </b>
 <a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self">Th1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:89_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E10:89_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> ) iff <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:89_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cat_3.html#D15" target="_self">Def15</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font> <a href="oppcat_1.html#K5">opp</a> 
;<br/>

<a NAME="E11:89_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="oppcat_1.html#K3">opp</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="cat_3.html#E9:89_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="oppcat_1.html#T13">OPPCAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E12:89_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="oppcat_1.html#T5">OPPCAT_1:5</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>;<br/>

<b>assume </b><a NAME="E13:89_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E14:89_1"/><b>then </b>
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T14">OPPCAT_1:14</a></i>;<br/>
<a NAME="E15:89_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T4">OPPCAT_1:4</a></i>;<br/>
<b>thus </b><a NAME="E16:89_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>8</sub></font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:89_1_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:89_1_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:89_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cat_3.html#E1:89_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cat_3.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E3:89_1_2_1"/><b>then </b>
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="cat_3.html#E1:89_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E4:89_1_2_1"/><b>then </b>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> 
 <b>by </b><i><a class="ref" href="oppcat_1.html#T11">OPPCAT_1:11</a></i>;<br/><a NAME="E5:89_1_2_1"/><b>then </b>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> 
 <b>by </b><i><a class="txt" href="cat_3.html#E1:89_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E6:89_1_2_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_3.html#E13:89_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><a NAME="E7:89_1_2_1"/>
 <a href="oppcat_1.html#K6">opp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="oppcat_1.html#K6">opp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_3.html#E1:89_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="cat_3.html#E1:89_1_2_1"><i><font color="Green">E58</font></i></a></i>;<br/><b>hence </b><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font> = 
 <a href="oppcat_1.html#K6">opp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E1:89_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E6:89_1_2_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="oppcat_1.html#T19">OPPCAT_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E1:89_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T7">OPPCAT_1:7</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E3:89_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="cat_3.html#E10:89_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="cat_3.html#E15:89_1"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:89_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="oppcat_1.html#T8">OPPCAT_1:8</a></i>;<br/>


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

<b>assume </b><a NAME="E17:89_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E18:89_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E19:89_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:89_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cat_3.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E20:89_1"/><b>then </b>
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="cat_3.html#E18:89_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E21:89_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_3.html#E15:89_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E22:89_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="cat_3.html#E17:89_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="oppcat_1.html#T7">OPPCAT_1:7</a></i>;<br/>
<a NAME="E23:89_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="cat_3.html#E10:89_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="cat_3.html#E15:89_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cat_3.html#E18:89_1"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E24:89_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> 
 
<b>by </b><i><a class="txt" href="cat_3.html#E21:89_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="oppcat_1.html#T17">OPPCAT_1:17</a></i>;<br/>
<a NAME="E25:89_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> 
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T8">OPPCAT_1:8</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>9</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> 
<b>by </b><i><a class="txt" href="cat_3.html#E18:89_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="oppcat_1.html#K5">opp</a> 
<b>by </b><i><a class="txt" href="cat_3.html#E18:89_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#D6" target="_self">Def6</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="oppcat_1.html#T8">OPPCAT_1:8</a></i>
;<br/><br/>


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

<b>assume </b><a NAME="E2:89"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a>  <a href="cat_3.html#R3" target="_self">is_a_coproduct_wrt</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> 
 ;<br/>

<a NAME="E3:89"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a>  is    <a href="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> ,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="cat_3.html#D18" target="_self">Def18</a></i>;<br/>
<b>hence </b><a NAME="E4:89"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="cat_3.html#T74" target="_self">Th74</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D15" target="_self">CAT_3:def 15</a><br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E5:89"/><i><font color="Green">E52</font></i>: 
 <a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="cat_3.html#K5" target="_self">opp</a>  as    <a href="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">c<sub>5</sub></font> <a href="oppcat_1.html#K3">opp</a> ,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="cat_3.html#T74" target="_self">Th74</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:89_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K3" target="_self">doms</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> = 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cat_3.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E1:89_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T9">OPPCAT_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="cat_3.html#E5:89"><i><font color="Green">E52</font></i></a>, <a class="txt" href="cat_3.html#E1:89_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cat_3.html#D4" target="_self">Def4</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E1:89_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cat_3.html#D4" target="_self">Def4</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T9">OPPCAT_1:9</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E1:89_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" target="_self">doms</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="cat_3.html#E1:89_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cat_3.html#D3" target="_self">Def3</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E8:89"/><b>then </b>
 <a href="cat_3.html#K3" target="_self">doms</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self">Th1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span><b> such that </b><br/><a NAME="E10:89"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>
 <b>and </b><br/><a NAME="E11:89"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K2">opp</a> </span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="oppcat_1.html#K3">opp</a> </span>)</span> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> ) iff <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E2:89"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cat_3.html#D18" target="_self">Def18</a></i>;<br/>
<b>take </b>
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font>
;<br/>

<a NAME="E12:89"/>
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="oppcat_1.html#K3">opp</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_3.html#E10:89"><i><font color="Green">E53</font></i></a>, <a class="ref" href="oppcat_1.html#T14">OPPCAT_1:14</a></i>;<br/>
<a NAME="E13:89"/><b>then </b>
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T4">OPPCAT_1:4</a></i>;<br/>
<b>hence </b><a NAME="E14:89"/>
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="oppcat_1.html#T4">OPPCAT_1:4</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E15:89"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E16:89"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T13">OPPCAT_1:13</a></i>;<br/>
<b>thus </b><a NAME="E17:89"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> ) implies  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:89_3"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:89_3_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:89_3_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a>  is    <a href="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> ,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cat_3.html#E2:89"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cat_3.html#D18" target="_self">Def18</a></i>;<br/><a NAME="E3:89_3_1"/><b>then </b>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> 
 <b>by </b><i><a class="txt" href="cat_3.html#E1:89_3_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#T67" target="_self">Th67</a></i>;<br/><a NAME="E4:89_3_1"/><b>then </b>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> 
 <b>by </b><i><a class="txt" href="cat_3.html#E1:89_3_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E5:89_3_1"/><b>then </b>
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> 
 <b>by </b><i><a class="ref" href="oppcat_1.html#T9">OPPCAT_1:9</a></i>;<br/><a NAME="E6:89_3_1"/><b>then </b>
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="oppcat_1.html#D2">OPPCAT_1:def 2</a></i>;<br/><a NAME="E7:89_3_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_3.html#E15:89"><i><font color="Green">E55</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><a NAME="E8:89_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="cat_3.html#E1:89_3"><i><font color="Green">E57</font></i></a>, <a class="txt" href="cat_3.html#E1:89_3_1"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E9:89_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> 
 <b>by </b><i><a class="txt" href="cat_3.html#E7:89_3_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="oppcat_1.html#T17">OPPCAT_1:17</a></i>;<br/><b>hence </b><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E1:89_3_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E1:89_3_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E3:89_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="oppcat_1.html#K5">opp</a> 
 
<b>by </b><i><a class="txt" href="cat_3.html#E11:89"><i><font color="Green">E54</font></i></a>, <a class="txt" href="cat_3.html#E16:89"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:89_3"/>
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="oppcat_1.html#T7">OPPCAT_1:7</a></i>;<br/>


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

<b>assume </b><a NAME="E18:89"/><i><font color="Green">E57</font></i>: 
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E19:89"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E20:89"/>
<font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a>  is    <a href="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> ,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="cat_3.html#E2:89"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cat_3.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E21:89"/><b>then </b>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> 
 
<b>by </b><i><a class="txt" href="cat_3.html#E19:89"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#T67" target="_self">Th67</a></i>;<br/>
<a NAME="E22:89"/><b>then </b>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> 
 
<b>by </b><i><a class="txt" href="cat_3.html#E19:89"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E23:89"/><b>then </b>
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K3">opp</a> 
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T9">OPPCAT_1:9</a></i>;<br/>
<a NAME="E24:89"/><b>then </b>
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#D2">OPPCAT_1:def 2</a></i>;<br/>
<a NAME="E25:89"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_3.html#E15:89"><i><font color="Green">E55</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E26:89"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="oppcat_1.html#K5">opp</a> 
 
<b>by </b><i><a class="txt" href="cat_3.html#E18:89"><i><font color="Green">E57</font></i></a>, <a class="ref" href="oppcat_1.html#T8">OPPCAT_1:8</a></i>;<br/>
<a NAME="E27:89"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="cat_3.html#E11:89"><i><font color="Green">E54</font></i></a>, <a class="txt" href="cat_3.html#E16:89"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cat_3.html#E19:89"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E28:89"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> 
 
<b>by </b><i><a class="txt" href="cat_3.html#E19:89"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="oppcat_1.html#D4">OPPCAT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> </span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E19:89"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="oppcat_1.html#K5">opp</a> 
<b>by </b><i><a class="txt" href="cat_3.html#E25:89"><i><font color="Green">E59</font></i></a>, <a class="ref" href="oppcat_1.html#T17">OPPCAT_1:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="oppcat_1.html#D4">OPPCAT_1:def 4</a></i>
;<br/><br/>


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