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<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">C</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">f</font>, <font color="Olive" title="b2">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">f</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b2">g</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">f</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">g</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">f</font>] )</span><br/>



<span class="kw">given </span><font color="Maroon" title="c7">fa</font>, <font color="Maroon" title="c8">fb</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c9">ff</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">fa</font>,<font color="Maroon" title="c8">fb</font><span class="kw"> such that </span><a NAME="E2:83_1"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c7">fa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E3:83_1"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<font color="Maroon" title="c8">fb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">and </span><br/><a NAME="E4:83_1"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">fa</font>,<font color="Maroon" title="c8">fb</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <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">and </span><br/><a NAME="E5:83_1"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
 for <font color="Olive" title="b1">o</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b2">h</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">fa</font> holds  <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">h</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b2">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon" title="c8">fb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"> (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">g</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">f</font>] )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c10">ga</font>, <font color="Maroon" title="c11">gb</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c12">gf</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">ga</font>,<font color="Maroon" title="c11">gb</font><span class="kw"> such that </span><a NAME="E7:83_1"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
<font color="Maroon" title="c10">ga</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">and </span><br/><a NAME="E8:83_1"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
<font color="Maroon" title="c11">gb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">and </span><br/><a NAME="E9:83_1"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">ga</font>,<font color="Maroon" title="c11">gb</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <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">and </span><br/><a NAME="E10:83_1"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
 for <font color="Olive" title="b1">o</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon" title="c10">ga</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b2">h</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c10">ga</font> holds  <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">h</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon" title="c10">ga</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">gf</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b2">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon" title="c11">gb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">f</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">gf</font> = <font color="Maroon" title="c12">gf</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">fb</font>,<font color="Maroon" title="c11">gb</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E7:83_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">fa</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">  ex <font color="Olive" title="b1">fb</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">g</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">fa</font>,<font color="Olive" title="b1">fb</font> st <br/>( <font color="Maroon" title="c7">fa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">fb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">fa</font>,<font color="Olive" title="b1">fb</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <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; (  for <font color="Olive" title="b3">o</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b3">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b4">h</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">o</font>,<font color="Maroon" title="c7">fa</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">h</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b4">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">o</font>,<font color="Olive" title="b1">fb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c11">gb</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">  ex <font color="Olive" title="b1">g</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">fa</font>,<font color="Maroon" title="c11">gb</font> st <br/>( <font color="Maroon" title="c7">fa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c11">gb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">fa</font>,<font color="Maroon" title="c11">gb</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <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; (  for <font color="Olive" title="b2">o</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b3">h</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">o</font>,<font color="Maroon" title="c7">fa</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">h</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b3">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">o</font>,<font color="Maroon" title="c11">gb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c14">k</font> = <font color="Maroon" title="c13">gf</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c9">ff</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="c7">fa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c11">gb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">fa</font>,<font color="Maroon" title="c11">gb</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <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; (  for <font color="Olive" title="b1">o</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b2">h</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">fa</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">h</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b2">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon" title="c11">gb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E12:83_1"/>
( <font color="Maroon" title="c7">fa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c11">gb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">fa</font>,<font color="Maroon" title="c11">gb</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <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="#E2:83_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E4:83_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E7:83_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E8:83_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E9:83_1"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="ref" href="altcat_1.html#D2" onmouseover="rs('altcat_1/D2')" onmouseout="rh()">ALTCAT_1:def 2</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">  for <font color="Olive" title="b1">o</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b2">h</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">fa</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">h</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b2">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon" title="c11">gb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c15">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</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"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  for <font color="Olive" title="b1">h</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">fa</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">h</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c11">gb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E13:83_1"/><span class="lab"><font color="Green" title="E46">A10</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">  for <font color="Olive" title="b1">h</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">fa</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">h</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c11">gb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<a NAME="E14:83_1"/><span class="lab"><font color="Green" title="E47">A11</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c8">fb</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <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="#E4:83_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E13:83_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="ref" href="altcat_1.html#D2" onmouseover="rs('altcat_1/D2')" onmouseout="rh()">ALTCAT_1:def 2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c16">h</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">fa</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"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">h</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c16">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c11">gb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<a NAME="E15:83_1"/><span class="lab"><font color="Green" title="E48">A12</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">h</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c16">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c8">fb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E13:83_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a></span>;<br/>
<a NAME="E16:83_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">h</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D2" onmouseover="rs('funct_1/D2')" onmouseout="rh()">FUNCT_1:def 2</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">h</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c7">fa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c16">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c8">fb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:83_1"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">gf</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">ff</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c16">h</font></span>)</span></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c11">gb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E7:83_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E10:83_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E14:83_1"><span class="lab"><font color="Green" title="E47">A11</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c16">h</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</font>,<font color="Maroon" title="c11">gb</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E4:83_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E7:83_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E9:83_1"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E13:83_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="ref" href="altcat_1.html#T21" onmouseover="rs('altcat_1/T21')" onmouseout="rh()">ALTCAT_1:21</a></span>
;<br/> <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/>


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