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<div class="add">

<b>let </b><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c3">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">C</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">I</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:91"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c4">F</font>
 <b>and </b><br/><a NAME="E2:91"/><i><font color="Green" title="E54">A2</font></i>: 
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">x1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">x2</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E3:91"/><i><font color="Green" title="E55">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">d</font> =  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> ) &amp; ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> implies $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> ) );<br/>
<a NAME="E4:91"/><i><font color="Green" title="E56">A4</font></i>: 
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="c1">I</font> holds <br/> ex <font color="Olive" title="b2">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:91_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<a NAME="E2:91_1"/><b>then </b><i><font color="Green" title="E57">A5</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="cat_3.html#E2:91"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="cat_3.html#E3:91"><i><font color="Green" title="E55">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>;<br/>
<a NAME="E4:91_1"/><i><font color="Green" title="E58">A6</font></i>: 
<font color="Maroon" title="c8">z</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">y</font></span>)</span>, <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cat_3.html#E2:91_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="cat_1.html#T21" title="CAT_1:th.21">CAT_1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1"/>
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> or <font color="Maroon" title="c7">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1"/><i><font color="Green" title="E59">A7</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c9">z</font> =  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>;<br/><b>thus </b><a NAME="E2:91_1_1_1"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">C</font>
 ;<br/><b>thus </b><a NAME="E3:91_1_1_1"/>
( ( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c7">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:91_1_1_1"><i><font color="Green" title="E59">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_2"/><i><font color="Green" title="E59">A8</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c8">z</font>
;<br/><b>thus </b><a NAME="E2:91_1_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="cat_3.html#E4:91_1"><i><font color="Green" title="E58">A6</font></i></a></i>;<br/><b>thus </b><a NAME="E3:91_1_1_2"/>
( ( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c7">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="cat_3.html#E2:91_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="cat_3.html#E1:91_1_1_2"><i><font color="Green" title="E59">A8</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">F'</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:91"/><i><font color="Green" title="E57">A9</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">F'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">C</font> )
 <b>and </b><br/><a NAME="E7:91"/><i><font color="Green" title="E58">A10</font></i>: 
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="c1">I</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c7">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="cat_3.html#E4:91"><i><font color="Green" title="E56">A4</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c8">F'</font> = <font color="Maroon" title="c7">F'</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="txt" href="cat_3.html#E6:91"><i><font color="Green" title="E57">A9</font></i></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#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:91_2_1"/><i><font color="Green" title="E59">A11</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E2:91_2_1"/><b>then </b>
( <font color="Maroon" title="c8">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c8">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E3:91"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="cat_3.html#D1" target="_self" title="CAT_3:def.1">Def1</a></i>;<br/><a NAME="E3:91_2_1"/><b>then </b>
( ( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c8">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c8">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="cat_3.html#E7:91"><i><font color="Green" title="E58">A10</font></i></a>, <a class="txt" href="cat_3.html#E1:91_2_1"><i><font color="Green" title="E59">A11</font></i></a></i>;<br/><a NAME="E4:91_2_1"/><b>then </b>
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a>, <a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c8">F'</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font> = 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E1:91_2_1"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="cat_3.html#D4" target="_self" title="CAT_3:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="txt" href="cat_3.html#E1:91_2_1"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="cat_3.html#T2" target="_self" title="CAT_3:th.2">Th2</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E2:91_2"/>
 <a href="cat_3.html#K4" title="CAT_3:func.4">cods</a> <font color="Maroon" title="c8">F'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c1">I</font> <a href="cat_3.html#K1" title="CAT_3:func.1">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self" title="CAT_3:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E3:91_2"/>
<font color="Maroon" title="c8">F'</font> is    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span>,<font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="cat_3.html#D17" target="_self" title="CAT_3:def.17">Def17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:91_2_2"/><i><font color="Green" title="E59">A12</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E2:91_2_2"/><b>then </b>
( <font color="Maroon" title="c8">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c8">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E3:91"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="cat_3.html#D1" target="_self" title="CAT_3:def.1">Def1</a></i>;<br/><a NAME="E3:91_2_2"/><b>then </b>
( ( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c8">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c9">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c8">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="cat_3.html#E7:91"><i><font color="Green" title="E58">A10</font></i></a>, <a class="txt" href="cat_3.html#E1:91_2_2"><i><font color="Green" title="E59">A12</font></i></a></i>;<br/><a NAME="E4:91_2_2"/><b>then </b>
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a>, <a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">doms</a> <font color="Maroon" title="c8">F'</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font> = 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E1:91_2_2"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="txt" href="cat_3.html#E1:91_2_2"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:91_2"/>
 <a href="cat_3.html#K3" title="CAT_3:func.3">doms</a> <font color="Maroon" title="c4">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K3" title="CAT_3:func.3">doms</a> <font color="Maroon" title="c8">F'</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self" title="CAT_3:th.1">Th1</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c9">i</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E11:91"/><i><font color="Green" title="E59">A13</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E12:91"/><i><font color="Green" title="E60">A14</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">c</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> 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="c1">I</font> holds <br/><font color="Olive" title="b1">k</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">y</font> ) iff <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:91"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="cat_3.html#D18" target="_self" title="CAT_3:def.18">Def18</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">i</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D11" target="_self" title="CAT_3:def.11">CAT_3:def 11</a><br/>

<b>thus </b> <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c9">i</font> = 
<font color="Maroon" title="c3">c</font>
<b>by </b><i><a class="txt" href="cat_3.html#E11:91"><i><font color="Green" title="E59">A13</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E3:91"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="cat_3.html#T67" target="_self" title="CAT_3:th.67">Th67</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c9">i</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span> = 
<font color="Maroon" title="c8">F'</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="cat_3.html#E3:91"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="cat_3.html#E11:91"><i><font color="Green" title="E59">A13</font></i></a>, <a class="txt" href="cat_3.html#E12:91"><i><font color="Green" title="E60">A14</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="cat_3.html#E3:91"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="cat_3.html#D1" target="_self" title="CAT_3:def.1">Def1</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E3:91"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="cat_3.html#E7:91"><i><font color="Green" title="E58">A10</font></i></a></i>
;<br/>


</div>
