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<b>let </b><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c4">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C</font>;<br/><b>let </b><font color="Maroon" title="c5">i1</font>, <font color="Maroon" title="c6">i2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font>;<br/>











<b>set </b><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font>;<br/>
<b>set </b><font color="Maroon" title="c8">I</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>
<b>assume </b><a NAME="E1:97"/><i><font color="Green" title="E57">A1</font></i>: 
<font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x2</font>
 ;<br/>

<b>thus </b><a NAME="E2:97"/>
( <font color="Maroon" title="c4">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font> implies <font color="Maroon" title="c4">c</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_1"/><i><font color="Green" title="E58">A2</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font>
 ;<br/>

<a NAME="E2:97_1"/><b>then </b>
(  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="ref" href="cat_3.html#D19" target="_self" title="CAT_3:def.19">Def19</a></i>;<br/>
<b>hence </b><a NAME="E3:97_1"/>
<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font> is    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c4">c</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="cat_3.html#T70" target="_self" title="CAT_3:th.70">Th70</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D18" target="_self" title="CAT_3:def.18">CAT_3:def 18</a><br/>

<b>let </b><font color="Maroon" title="c9">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C</font>;<br/><b>let </b><font color="Maroon" title="c10">F'</font> be    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c9">b</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>



<b>assume </b><a NAME="E4:97_1"/><i><font color="Green" title="E59">A3</font></i>: 
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c10">F'</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font>;<br/>
<b>set </b><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font>;<br/>
<a NAME="E5:97_1"/><i><font color="Green" title="E60">A4</font></i>: 
( <font color="Maroon" title="c1">x1</font> <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"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c2">x2</font> <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"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E6:97_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span> &amp; <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="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></i>;<br/>
<a NAME="E7:97_1"/><b>then </b>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E4:97_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="cat_3.html#E5:97_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></i>;<br/>
<a NAME="E8:97_1"/><b>then </b>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">i1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">i2</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:97"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="cat_3.html#E5:97_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="cat_3.html#T7" target="_self" title="CAT_3:th.7">Th7</a>, <a class="ref" href="cat_3.html#T67" target="_self" title="CAT_3:th.67">Th67</a></i>;<br/>
<a NAME="E9:97_1"/><b>then </b>
( <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">i1</font></span>)</span>,<font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">i2</font></span>)</span>,<font color="Maroon" title="c9">b</font> )
 
;<br/>
<b>then consider </b><font color="Maroon" title="c13">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font><b> such that </b><br/><a NAME="E11:97_1"/><i><font color="Green" title="E61">A5</font></i>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>
 <b>and </b><br/><a NAME="E12:97_1"/><i><font color="Green" title="E62">A6</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="c3">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#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font> holds <br/>( ( <font color="Olive" title="b1">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Olive" title="b1">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font> ) iff <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:97_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="cat_3.html#D19" target="_self" title="CAT_3:def.19">Def19</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">h</font>
;<br/>

<b>thus </b><a NAME="E13:97_1"/>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="cat_3.html#E11:97_1"><i><font color="Green" title="E61">A5</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c14">k</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font>;<br/>

<b>assume </b><a NAME="E14:97_1"/><i><font color="Green" title="E63">A7</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>
 ;<br/>

<b>thus </b><a NAME="E15:97_1"/>
( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> ) implies <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font>
 ;<br/>

<a NAME="E2:97_1_1"/><b>then </b>
( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E5:97_1"><i><font color="Green" title="E60">A4</font></i></a></i>;<br/>
<a NAME="E3:97_1_1"/><b>then </b>
( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:97"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="cat_3.html#T7" target="_self" title="CAT_3:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E4:97_1_1"/>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font>
 <b>by </b><i><a class="txt" href="cat_3.html#E12:97_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="cat_3.html#E14:97_1"><i><font color="Green" title="E63">A7</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E16:97_1"/><i><font color="Green" title="E64">A8</font></i>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E17:97_1"/><i><font color="Green" title="E65">A9</font></i>: 
<font color="Maroon" title="c15">x</font> <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"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/>

<a NAME="E18:97_1"/>
( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E12:97_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="cat_3.html#E14:97_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="cat_3.html#E16:97_1"><i><font color="Green" title="E64">A8</font></i></a></i>;<br/>
<a NAME="E19:97_1"/><b>then </b>
( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font> &amp; ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> ) )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:97"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="cat_3.html#E17:97_1"><i><font color="Green" title="E65">A9</font></i></a>, <a class="ref" href="cat_3.html#T7" target="_self" title="CAT_3:th.7">Th7</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E20:97_1"/>
<font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font>
 ;<br/>


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

<b>assume </b><a NAME="E3:97"/><i><font color="Green" title="E58">A10</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font>
 ;<br/>

<a NAME="E4:97"/><b>then </b>
( <font color="Maroon" title="c1">x1</font> <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"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c2">x2</font> <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"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font> is    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c4">c</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="ref" href="cat_3.html#D18" target="_self" title="CAT_3:def.18">Def18</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:97"/><b>then </b>
(  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="ref" href="cat_3.html#T67" target="_self" title="CAT_3:th.67">Th67</a></i>;<br/>
<b>hence </b><a NAME="E6:97"/>
(  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:97"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="cat_3.html#T7" target="_self" title="CAT_3:th.7">Th7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D19" target="_self" title="CAT_3:def.19">CAT_3:def 19</a><br/>

<b>let </b><font color="Maroon" title="c9">d</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C</font>;<br/><b>let </b><font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c11">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E7:97"/><i><font color="Green" title="E59">A11</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">i1</font></span>)</span>,<font color="Maroon" title="c9">d</font>
 <b>and </b><br/><a NAME="E8:97"/><i><font color="Green" title="E60">A12</font></i>: 
<font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">i2</font></span>)</span>,<font color="Maroon" title="c9">d</font>
 ;<br/>

<a NAME="E9:97"/>
(  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E7:97"><i><font color="Green" title="E59">A11</font></i></a>, <a class="txt" href="cat_3.html#E8:97"><i><font color="Green" title="E60">A12</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">F'</font> = <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font> as    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c9">d</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <b>by </b><i><a class="ref" href="cat_3.html#T70" target="_self" title="CAT_3:th.70">Th70</a></i>;<br/>
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> = 
<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">i1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">i2</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_3.html#T10" target="_self" title="CAT_3:th.10">Th10</a></i>
<br/>.= 
<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c10">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">i2</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E7:97"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>
<br/>.= 
<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c10">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c11">g</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E8:97"><i><font color="Green" title="E60">A12</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_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c12">F'</font>
<b>by </b><i><a class="ref" href="cat_3.html#T10" target="_self" title="CAT_3:th.10">Th10</a></i>
;<br/>
<b>then consider </b><font color="Maroon" title="c13">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font><b> such that </b><br/><a NAME="E13:97"/><i><font color="Green" title="E61">A13</font></i>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>
 <b>and </b><br/><a NAME="E14:97"/><i><font color="Green" title="E62">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="c3">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#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font> holds <br/>( ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><font color="Olive" title="b1">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font> ) iff <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E3:97"><i><font color="Green" title="E58">A10</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="c13">h</font>
;<br/>

<b>thus </b><a NAME="E15:97"/>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>
 <b>by </b><i><a class="txt" href="cat_3.html#E13:97"><i><font color="Green" title="E61">A13</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c14">k</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font>;<br/>

<b>assume </b><a NAME="E16:97"/><i><font color="Green" title="E63">A15</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>
 ;<br/>

<b>thus </b><a NAME="E17:97"/>
( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> implies <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_3"/><i><font color="Green" title="E64">A16</font></i>: 
( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:97_3_1"/>
<font color="Maroon" title="c15">x</font> <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"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/><a NAME="E2:97_3_1"/><b>then </b>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:97_3_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i1</font> &amp; <font color="Maroon" title="c12">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> ) or ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c12">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> ) )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:97"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="cat_3.html#T7" target="_self" title="CAT_3:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E4:97_3_1"/>
<font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="cat_3.html#E1:97_3"><i><font color="Green" title="E64">A16</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:97_3"/>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font>
 <b>by </b><i><a class="txt" href="cat_3.html#E14:97"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="cat_3.html#E16:97"><i><font color="Green" title="E63">A15</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E18:97"/>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font>
 ;<br/>

<a NAME="E19:97"/><b>then </b>
( <font color="Maroon" title="c1">x1</font> <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"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c2">x2</font> <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"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; ( 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> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> ) )
 
<b>by </b><i><a class="txt" href="cat_3.html#E14:97"><i><font color="Green" title="E62">A14</font></i></a>, <a class="txt" href="cat_3.html#E16:97"><i><font color="Green" title="E63">A15</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E20:97"/><b>then </b>
( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font> )
 
;<br/>
<a NAME="E21:97"/><b>then </b>
( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">i2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E1:97"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="cat_3.html#T7" target="_self" title="CAT_3:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E22:97"/>
( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:97"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="cat_3.html#T7" target="_self" title="CAT_3:th.7">Th7</a></i>;<br/>


</div>
