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<a NAME="E1:68_1_1"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
( <font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c4">c</font>, <a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>), <a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>) <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c4">c</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T42" target="_self" onmouseover="rs('cat_8/T42')" onmouseout="rh()">Th42</a></span>;<br/>
<a NAME="E2:68_1_1"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
( <font color="Maroon" title="c3">b</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c5">d</font>, <a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>), <a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>) <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c5">d</font></span>)</span>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c5">d</font></span>)</span>,<font color="Maroon" title="c5">d</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T42" target="_self" onmouseover="rs('cat_8/T42')" onmouseout="rh()">Th42</a></span>;<br/>
<a NAME="E3:68_1_1"/>
(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c4">c</font></span>)</span>,<font color="Maroon" title="c5">d</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1_1"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="cat_7.html#T22" onmouseover="rs('cat_7/T22')" onmouseout="rh()">CAT_7:22</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c5">d</font><span class="kw"> such that </span><br/><a NAME="E5:68_1_1"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
( <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c5">d</font>  st <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> holds <br/><font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="cat_8.html#D10" target="_self" onmouseover="rs('cat_8/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E6:68_1_1"/>
 for <font color="Olive" title="b1">h1</font>, <font color="Olive" title="b2">h2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c5">d</font>  st <font color="Maroon" title="c6">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> &amp; <font color="Maroon" title="c7">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> &amp; <font color="Maroon" title="c6">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h2</font> &amp; <font color="Maroon" title="c7">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h2</font> holds <br/><font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">h1</font>, <font color="Maroon" title="c10">h2</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c5">d</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="c6">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">h1</font> &amp; <font color="Maroon" title="c7">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">h1</font> &amp; <font color="Maroon" title="c6">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">h2</font> &amp; <font color="Maroon" title="c7">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">h2</font> implies <font color="Maroon" title="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:68_1_1_1"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
( <font color="Maroon" title="c6">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">h1</font> &amp; <font color="Maroon" title="c7">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">h1</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 <font color="Maroon" title="c6">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">h2</font> or  not <font color="Maroon" title="c7">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">h2</font> or <font color="Maroon" title="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:68_1_1_1"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
( <font color="Maroon" title="c6">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">h2</font> &amp; <font color="Maroon" title="c7">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">h2</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="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h2</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c9">h1</font> = 
<font color="Maroon" title="c8">h</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1_1_1"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="txt" href="#E5:68_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">h2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E2:68_1_1_1"><span class="lab"><font color="Green" title="E42">A7</font></span></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="E7:68_1_1"/>
(  ex <font color="Olive">b<sub>1</sub></font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c5">d</font> st <br/>( <font color="Maroon" title="c6">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c7">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive">b<sub>1</sub></font> ) &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c5">d</font>  st <font color="Maroon" title="c6">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c7">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c6">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon" title="c7">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive">b<sub>2</sub></font> holds <br/><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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></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>
