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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C</font> being   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><br/>  for <font color="Olive" title="b2">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Olive" title="b1">C</font><br/>  for <font color="Olive" title="b3">i1</font>, <font color="Olive" title="b4">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">C</font>  st <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> holds <br/>( <font color="Olive" title="b2">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b3">i1</font>,<font color="Olive" title="b4">i2</font> iff <font color="Olive" title="b2">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="Olive" title="b3">i1</font>,<font color="Olive" title="b4">i2</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C</font><br/>  for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font>  st <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> holds <br/>( <font color="Olive" title="b1">c</font> <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font> iff <font color="Olive" title="b1">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="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font>) )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font>  st <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> holds <br/>( <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="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> iff <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="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font>) )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x2</font> implies ( <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> iff <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>) ) )</span><br/>











<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">assume </span><a NAME="E1:96"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> iff <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>) )</span><br/>

<span class="kw">thus </span><a NAME="E2:96"/>
( <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>) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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>) implies <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:96_1"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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>)</span><br/>

<a NAME="E2:96_1"/><span class="kw">then </span>
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">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="graph_1.html#K2" title="GRAPH_1:func.2">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D17" target="_self" title="CAT_3:def.17">Def19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:96_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T65" target="_self" title="CAT_3:th.65">Th70</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_3.html#D16" target="_self" title="CAT_3:def.16">CAT_3:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">d</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C</font><br/>  for <font color="Olive" title="b2">F9</font> being    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive" title="b1">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>  st  <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="Olive" title="b2">F9</font> holds <br/> ex <font color="Olive" title="b3">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font> st <br/>( <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font>) &amp; (  for <font color="Olive" title="b4">k</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font>) holds <br/>( (  for <font color="Olive" title="b5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="b4">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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="b5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">x</font> ) iff <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">k</font> ) ) )</span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F9</font> being    <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>  st  <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="Olive" title="b1">F9</font> holds <br/> ex <font color="Olive" title="b2">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font> st <br/>( <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>) &amp; (  for <font color="Olive" title="b3">k</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>) holds <br/>( (  for <font color="Olive" title="b4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="b3">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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="b4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">x</font> ) iff <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c10">F9</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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">F9</font> implies  ex <font color="Olive" title="b1">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>) &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>) holds <br/>( (  for <font color="Olive" title="b3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="b2">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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="b3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x</font> ) iff <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> ) ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E4:96_1"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
 <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">F9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>) &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>) holds <br/>( (  for <font color="Olive" title="b3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="b2">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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="b3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x</font> ) iff <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> ) ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">x1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">x2</font>;<br/>
<a NAME="E5:96_1"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E6:96_1"/><span class="kw">then </span>
<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="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><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="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D1" target="_self" title="CAT_3:def.1">Def3</a></span>;<br/>
<a NAME="E7:96_1"/><span class="kw">then </span>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><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="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E5:96_1"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="cat_3.html#D1" target="_self" title="CAT_3:def.1">Def3</a></span>;<br/>
<a NAME="E8:96_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A5</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="cat_3.html#T3" target="_self" title="CAT_3:th.3">Th7</a></span>;<br/>
<a NAME="E9:96_1"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E10:96_1"/><span class="kw">then </span>
<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="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><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="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D1" target="_self" title="CAT_3:def.1">Def3</a></span>;<br/>
<a NAME="E11:96_1"/><span class="kw">then </span>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><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="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E9:96_1"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="ref" href="cat_3.html#D1" target="_self" title="CAT_3:def.1">Def3</a></span>;<br/>
<a NAME="E12:96_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A7</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="cat_3.html#T3" target="_self" title="CAT_3:th.3">Th7</a></span>;<br/>
<a NAME="E13:96_1"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:96_1"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="ref" href="cat_3.html#T62" target="_self" title="CAT_3:th.62">Th67</a></span>;<br/>
<a NAME="E14:96_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A8</font></span>: 
<font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">i2</font></span>)</span>,<font color="Maroon" title="c9">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:96_1"><span class="lab"><font color="Green" title="E62">A7</font></span></a></span>;<br/>
<a NAME="E15:96_1"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_1"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="cat_3.html#T62" target="_self" title="CAT_3:th.62">Th67</a></span>;<br/>
<a NAME="E16:96_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">i1</font></span>)</span>,<font color="Maroon" title="c9">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:96_1"><span class="lab"><font color="Green" title="E60">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E18:96_1"/><span class="lab"><font color="Green" title="E64">A9</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>)
 <span class="kw">and </span><br/><a NAME="E19:96_1"/><span class="lab"><font color="Green" title="E65">A10</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="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#K3" title="CAT_1:func.3">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#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E14:96_1"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="ref" href="cat_3.html#D17" target="_self" title="CAT_3:def.17">Def19</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">h</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>) &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="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#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</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#K1" title="CAT_1:func.1">*</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="c10">F9</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> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E20:96_1"/>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:96_1"><span class="lab"><font color="Green" title="E64">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" 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#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</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#K1" title="CAT_1:func.1">*</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="c10">F9</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> )</span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>) implies ( (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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#K1" title="CAT_1:func.1">*</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">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> ) iff <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E21:96_1"/><span class="lab"><font color="Green" title="E66">A11</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">b</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" 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#K1" title="CAT_1:func.1">*</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">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> ) iff <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> )</span><br/>

<span class="kw">thus </span><a NAME="E22:96_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#K1" title="CAT_1:func.1">*</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> implies  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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#K1" title="CAT_1:func.1">*</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">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:96_1_1"/><span class="lab"><font color="Green" title="E67">A12</font></span>: 
 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#K1" title="CAT_1:func.1">*</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font></span><br/>

<a NAME="E2:96_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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#K10" title="FUNCT_2:func.10">/.</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">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:96_1"><span class="lab"><font color="Green" title="E61">A6</font></span></a></span>;<br/>
<a NAME="E3:96_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A13</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="cat_3.html#T3" target="_self" title="CAT_3:th.3">Th7</a></span>;<br/>
<a NAME="E4:96_1_1"/>
<font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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#K10" title="FUNCT_2:func.10">/.</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">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_1"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E1:96_1_1"><span class="lab"><font color="Green" title="E67">A12</font></span></a></span>;<br/>
<a NAME="E5:96_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="cat_3.html#T3" target="_self" title="CAT_3:th.3">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:96_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:96_1"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="txt" href="#E21:96_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="txt" href="#E3:96_1_1"><span class="lab"><font color="Green" title="E68">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E23:96_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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#K1" title="CAT_1:func.1">*</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">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span><br/>

<a NAME="E24:96_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A14</font></span>: 
( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">x1</font> &amp; <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">x2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:96_1"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="txt" href="#E21:96_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> implies <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E25:96_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font></span><br/>

<a NAME="E26:96_1"/><span class="kw">then </span>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:96_1"/>
<font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E24:96_1"><span class="lab"><font color="Green" title="E67">A14</font></span></a>, <a class="ref" href="cat_3.html#T3" target="_self" title="CAT_3:th.3">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E3:96"/><span class="lab"><font color="Green" title="E57">A15</font></span>: 
<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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/>

<a NAME="E4:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A16</font></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> (<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D16" target="_self" title="CAT_3:def.16">Def18</a></span>;<br/>
<a NAME="E5:96"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E6:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A17</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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#K10" title="FUNCT_2:func.10">/.</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="cat_3.html#T62" target="_self" title="CAT_3:th.62">Th67</a></span>;<br/>
<a NAME="E7:96"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E8:96"/><span class="kw">then </span>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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#K10" title="FUNCT_2:func.10">/.</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="cat_3.html#T62" target="_self" title="CAT_3:th.62">Th67</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:96"/>
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">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="graph_1.html#K2" title="GRAPH_1:func.2">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E6:96"><span class="lab"><font color="Green" title="E59">A17</font></span></a>, <a class="ref" href="cat_3.html#T3" target="_self" title="CAT_3:th.3">Th7</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_3.html#D17" target="_self" title="CAT_3:def.17">CAT_3:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">d</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C</font><br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</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="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">i1</font></span>)</span>,<font color="Olive" title="b1">d</font>) &amp; <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">i2</font></span>)</span>,<font color="Olive" title="b1">d</font>) holds <br/> ex <font color="Olive" title="b4">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font> st <br/>( <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font>) &amp; (  for <font color="Olive" title="b5">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="b5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">d</font>) holds <br/>( ( <font color="Olive" title="b5">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b5">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) iff <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">k</font> ) ) )</span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</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">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">i1</font></span>)</span>,<font color="Maroon" title="c9">d</font>) &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">i2</font></span>)</span>,<font color="Maroon" title="c9">d</font>) holds <br/> ex <font color="Olive" title="b3">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font> st <br/>( <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>) &amp; (  for <font color="Olive" title="b4">k</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>) holds <br/>( ( <font color="Olive" title="b4">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b4">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) iff <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">k</font> ) ) )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">i1</font></span>)</span>,<font color="Maroon" title="c9">d</font>) &amp; <font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">i2</font></span>)</span>,<font color="Maroon" title="c9">d</font>) implies  ex <font color="Olive" title="b1">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>) &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>) holds <br/>( ( <font color="Olive" title="b2">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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="Olive" title="b2">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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> ) iff <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> ) ) ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E10:96"/><span class="lab"><font color="Green" title="E60">A18</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">i1</font></span>)</span>,<font color="Maroon" title="c9">d</font>)
 <span class="kw">and </span><br/><a NAME="E11:96"/><span class="lab"><font color="Green" title="E61">A19</font></span>: 
<font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">i2</font></span>)</span>,<font color="Maroon" title="c9">d</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>) &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>) holds <br/>( ( <font color="Olive" title="b2">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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="Olive" title="b2">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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> ) iff <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> ) ) )</span><br/>

<a NAME="E12:96"/>
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">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="graph_1.html#K2" title="GRAPH_1:func.2">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:96"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E11:96"><span class="lab"><font color="Green" title="E61">A19</font></span></a>, <a class="ref" href="cat_1.html#T1" title="CAT_1:th.1">CAT_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">F9</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T65" target="_self" title="CAT_3:th.65">Th70</a></span>;<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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">i1</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">i2</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T6" target="_self" title="CAT_3:th.6">Th10</a></span>
<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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">i2</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:96"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="cat_1.html#T1" title="CAT_1:th.1">CAT_1:1</a></span>
<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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">g</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:96"><span class="lab"><font color="Green" title="E61">A19</font></span></a>, <a class="ref" href="cat_1.html#T1" title="CAT_1:th.1">CAT_1:1</a></span>
<br/>.= 
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c12">F9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T6" target="_self" title="CAT_3:th.6">Th10</a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E16:96"/><span class="lab"><font color="Green" title="E62">A20</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>)
 <span class="kw">and </span><br/><a NAME="E17:96"/><span class="lab"><font color="Green" title="E63">A21</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="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#K3" title="CAT_1:func.3">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#K1" title="CAT_1:func.1">*</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="ref" href="cat_3.html#D16" target="_self" title="CAT_3:def.16">Def18</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">h</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>) &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <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#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>) holds <br/>( ( <font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</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> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E18:96"/>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:96"><span class="lab"><font color="Green" title="E62">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" 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#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>) holds <br/>( ( <font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</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> )</span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>) implies ( ( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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#K1" title="CAT_1:func.1">*</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> ) iff <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E19:96"/><span class="lab"><font color="Green" title="E64">A22</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c9">d</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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#K1" title="CAT_1:func.1">*</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> ) iff <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> )</span><br/>

<span class="kw">thus </span><a NAME="E20:96"/>
( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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#K1" title="CAT_1:func.1">*</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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> implies ( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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#K1" title="CAT_1:func.1">*</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> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:96_3"/><span class="lab"><font color="Green" title="E65">A23</font></span>: 
( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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#K1" title="CAT_1:func.1">*</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> implies <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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#K10" title="FUNCT_2:func.10">/.</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">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c15">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:96_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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#K10" title="FUNCT_2:func.10">/.</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">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c15">x</font></span><br/><a NAME="E2:96_3_1"/><span class="kw">then </span>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E3:96_3_1"/><span class="kw">then </span>
( ( <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#K10" title="FUNCT_2:func.10">/.</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">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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="funct_2.html#K10" title="FUNCT_2:func.10">/.</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">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="cat_3.html#T3" target="_self" title="CAT_3:th.3">Th7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:96_3_1"/>
<font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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#K10" title="FUNCT_2:func.10">/.</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">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_3"><span class="lab"><font color="Green" title="E65">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:96_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:96"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="txt" href="#E19:96"><span class="lab"><font color="Green" title="E64">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E21:96"/><span class="lab"><font color="Green" title="E65">A24</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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#K1" title="CAT_1:func.1">*</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> )</span><br/>

<a NAME="E22:96"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E23:96"/><span class="kw">then </span>
<font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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#K10" title="FUNCT_2:func.10">/.</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">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c2">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:96"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="txt" href="#E19:96"><span class="lab"><font color="Green" title="E64">A22</font></span></a>, <a class="txt" href="#E21:96"><span class="lab"><font color="Green" title="E65">A24</font></span></a></span>;<br/>
<a NAME="E24:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A25</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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#K10" title="FUNCT_2:func.10">/.</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="cat_3.html#T3" target="_self" title="CAT_3:th.3">Th7</a></span>;<br/>
<a NAME="E25:96"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E26:96"/><span class="kw">then </span>
<font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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#K10" title="FUNCT_2:func.10">/.</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">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c1">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:96"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="txt" href="#E19:96"><span class="lab"><font color="Green" title="E64">A22</font></span></a>, <a class="txt" href="#E21:96"><span class="lab"><font color="Green" title="E65">A24</font></span></a></span>;<br/>
<a NAME="E27:96"/><span class="kw">then </span>
<font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="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#K10" title="FUNCT_2:func.10">/.</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="cat_3.html#T3" target="_self" title="CAT_3:th.3">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:96"/>
( <font color="Maroon" title="c14">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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#K1" title="CAT_1:func.1">*</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E24:96"><span class="lab"><font color="Green" title="E66">A25</font></span></a>, <a class="ref" href="cat_3.html#T3" target="_self" title="CAT_3:th.3">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
