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<b>let </b><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">f</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/><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="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font></span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Olive" title="b4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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> <span class="p1">(<span class="default"><font color="Olive" title="b2">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b4">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b4">f</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">f</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/><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="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font></span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Olive" title="b3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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> <span class="p1">(<span class="default"><font color="Olive" title="b1">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b3">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b3">f</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">p1</font>, <font color="Maroon" title="c5">p2</font>, <font color="Maroon" title="c6">f</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:()"><i><font color="Red">::  thesis: </font></i></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 <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="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span> )</span><br/>











<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>;<br/>
<b>set </b><font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">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 <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="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> )</span><br/><b>assume </b><a NAME="E1:22_1"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c9">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font></span><br/><a NAME="E2:22_1"/><b>then </b>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:22_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font> <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p1</font> &amp; <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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">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="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p2</font> &amp; <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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font> ) )
 <b>by </b><i><a class="txt" href="cat_3.html#E1:22"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="cat_3.html#T7" target="_self" title="CAT_3:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E4:22_1"/>
<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="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="cat_3.html#E1:22_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="cat_3.html#D7" target="_self" title="CAT_3:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:22"/>
<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="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self" title="CAT_3:th.1">Th1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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