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<a NAME="E1:87_1_1"/>
(  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  implies (  ex <font color="Olive" title="b1">f</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span> st <br/>( <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> ) &amp; (  for <font color="Olive" title="b1">f11</font>, <font color="Olive" title="b2">f22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span>  st <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> holds <br/><font color="Olive" title="b1">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f22</font> ) ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:87_1_1_1"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
(  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">f</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span> st <br/>( <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> ) &amp; (  for <font color="Olive" title="b1">f11</font>, <font color="Olive" title="b2">f22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span>  st <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> holds <br/><font color="Olive" title="b1">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f22</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">D1</font> = <font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c6">D2</font> = <font color="Maroon" title="c2">C2</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_1_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">g1</font> = <font color="Maroon" title="c3">f1</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c5">D1</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">g2</font> = <font color="Maroon" title="c4">f2</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c6">D2</font> ;<br/>
<a NAME="E5:87_1_1_1"/>
<font color="Maroon" title="c5">D1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c6">D2</font>, <a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>), <a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>) <a href="cat_8.html#R2" title="CAT_8:pred.2">is_product_of</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T48" target="_self" onmouseover="rs('cat_8/T48')" onmouseout="rh()">Th48</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c6">D2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:87_1_1_1"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
( <font color="Maroon" title="c9">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c7">g1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c8">g2</font> &amp; (  for <font color="Olive" title="b1">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c6">D2</font></span>)</span>  st <font color="Olive" title="b1">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c7">g1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Olive" title="b1">H1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c8">g2</font> holds <br/><font color="Maroon" title="c9">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">H1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#D17" target="_self" onmouseover="rs('cat_8/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">g12</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E9:87_1_1_1"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
(  not <font color="Maroon" title="c10">g12</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp;  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c10">g12</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c10">g12</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp;  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <span class="p1">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c10">g12</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c10">g12</font></span>)</span>,<font color="Maroon" title="c10">g12</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> &amp;  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c10">g12</font>, <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c10">g12</font>,<font color="Maroon" title="c10">g12</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T39" onmouseover="rs('cat_7/T39')" onmouseout="rh()">CAT_7:39</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c9">H</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span> ;<br/>
<span class="kw">thus </span><a NAME="E11:87_1_1_1"/>
 ex <font color="Olive" title="b1">f</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span> st <br/>( <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f11</font>, <font color="Olive" title="b2">f22</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span>  st <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> holds <br/><font color="Olive" title="b1">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f22</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c11">f</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> )</span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_1_1_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c3">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_1_1_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_7.html#D16" onmouseover="rs('cat_7/D16')" onmouseout="rh()">CAT_7:def 16</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c9">H</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_1_1_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<font color="Maroon" title="c4">f2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_1_1_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_7.html#D16" onmouseover="rs('cat_7/D16')" onmouseout="rh()">CAT_7:def 16</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c12">f11</font>, <font color="Maroon" title="c13">f22</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> implies <font color="Maroon" title="c12">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f22</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">g11</font> = <font color="Maroon" title="c12">f11</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c6">D2</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">g22</font> = <font color="Maroon" title="c13">f22</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c6">D2</font></span>)</span> ;<br/>
<span class="kw">assume </span><a NAME="E14:87_1_1_1"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
( <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c12">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> or  not <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> or <font color="Maroon" title="c12">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f22</font> )</span><br/>

<span class="kw">assume </span><a NAME="E15:87_1_1_1"/><span class="lab"><font color="Green" title="E49">A5</font></span>: 
( <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c13">f22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f22</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c16">H1</font> =  <a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c14">g11</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">H2</font> =  <a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c15">g22</font>;<br/>
<a NAME="E16:87_1_1_1"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
<span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c14">g11</font></span>)</span> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c14">g11</font></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font> = 
<span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c14">g11</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">g11</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_7.html#D16" onmouseover="rs('cat_7/D16')" onmouseout="rh()">CAT_7:def 16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:87_1_1_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_7.html#D16" onmouseover="rs('cat_7/D16')" onmouseout="rh()">CAT_7:def 16</a></span>
;<br/>
<a NAME="E18:87_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A7</font></span>: 
<span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c14">g11</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c7">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:87_1_1_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_8.html#T17" target="_self" onmouseover="rs('cat_8/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E19:87_1_1_1"/><span class="lab"><font color="Green" title="E52">A8</font></span>: 
<span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c14">g11</font></span>)</span> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c14">g11</font></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font> = 
<span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c14">g11</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c14">g11</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_7.html#D16" onmouseover="rs('cat_7/D16')" onmouseout="rh()">CAT_7:def 16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c8">g2</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:87_1_1_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_7.html#D16" onmouseover="rs('cat_7/D16')" onmouseout="rh()">CAT_7:def 16</a></span>
;<br/>
<a NAME="E21:87_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c14">g11</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c8">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:87_1_1_1"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_8.html#T17" target="_self" onmouseover="rs('cat_8/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E22:87_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A9</font></span>: 
 <a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c14">g11</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:87_1_1_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E7:87_1_1_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a></span>;<br/>
<a NAME="E23:87_1_1_1"/><span class="lab"><font color="Green" title="E54">A10</font></span>: 
<span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c15">g22</font></span>)</span> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c15">g22</font></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font> = 
<span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c15">g22</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">g22</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_7.html#D16" onmouseover="rs('cat_7/D16')" onmouseout="rh()">CAT_7:def 16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c7">g1</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:87_1_1_1"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_7.html#D16" onmouseover="rs('cat_7/D16')" onmouseout="rh()">CAT_7:def 16</a></span>
;<br/>
<a NAME="E25:87_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A11</font></span>: 
<span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c15">g22</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c7">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:87_1_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_8.html#T17" target="_self" onmouseover="rs('cat_8/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E26:87_1_1_1"/><span class="lab"><font color="Green" title="E56">A12</font></span>: 
<span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c15">g22</font></span>)</span> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T35" onmouseover="rs('cat_6/T35')" onmouseout="rh()">CAT_6:35</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p2">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c15">g22</font></span>)</span></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font> = 
<span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c15">g22</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T34" onmouseover="rs('cat_6/T34')" onmouseout="rh()">CAT_6:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c15">g22</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_7.html#D16" onmouseover="rs('cat_7/D16')" onmouseout="rh()">CAT_7:def 16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c8">g2</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">g12</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:87_1_1_1"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_7.html#D16" onmouseover="rs('cat_7/D16')" onmouseout="rh()">CAT_7:def 16</a></span>
;<br/>
<a NAME="E28:87_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c6">D2</font>)</span>)</span> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c15">g22</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_7.html#K7" title="CAT_7:func.7">MORPHISM</a> <font color="Maroon" title="c8">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:87_1_1_1"><span class="lab"><font color="Green" title="E56">A12</font></span></a>, <a class="txt" href="#E9:87_1_1_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="cat_8.html#T17" target="_self" onmouseover="rs('cat_8/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:87_1_1_1"/>
<font color="Maroon" title="c12">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f22</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_1_1_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E25:87_1_1_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="txt" href="#E22:87_1_1_1"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="ref" href="cat_8.html#T16" target="_self" onmouseover="rs('cat_8/T16')" onmouseout="rh()">Th16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:87_1_1"/>
( (  for <font color="Olive">b<sub>1</sub></font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span> holds  verum ) &amp; (  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span> st <br/>( <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> ) ) &amp; ( ( <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  ) implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span> ) &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span> holds <br/> ( (  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K9" title="CAT_8:func.9">pr1</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K10" title="CAT_8:func.10">pr2</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( ( <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  ) &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C1</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c2">C2</font></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
