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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b3">F1</font>, <font color="Olive" title="b4">F2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b3">F1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) &amp; <font color="Olive" title="b4">F2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) &amp; <font color="Olive" title="b3">F1</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"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</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> <font color="Olive" title="b4">F2</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"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</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> holds <br/><font color="Olive" title="b3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">F2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</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">F1</font>, <font color="Olive" title="b2">F2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">F1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) &amp; <font color="Olive" title="b2">F2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) &amp; <font color="Olive" title="b1">F1</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"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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> <font color="Olive" title="b2">F2</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"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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> holds <br/><font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F2</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">B</font> =  <a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">F1</font>, <font color="Maroon" title="c6">F2</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" title="c5">F1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) &amp; <font color="Maroon" title="c6">F2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) &amp; <font color="Maroon" title="c5">F1</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"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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> <font color="Maroon" title="c6">F2</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"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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> implies <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:87"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<font color="Maroon" title="c5">F1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)
 <span class="kw">and </span><br/><a NAME="E2:87"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c6">F2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)
 <span class="kw">and </span><br/><a NAME="E3:87"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c5">F1</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"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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> <font color="Maroon" title="c6">F2</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"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font></span><br/>

<a NAME="E4:87"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c5">F1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yellow18.html#K3" title="YELLOW18:func.3">-carrier_of</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yellow18.html#K3" title="YELLOW18:func.3">-carrier_of</a> <font color="Maroon" title="c3">b</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="yellow18.html#D12" target="_self" onmouseover="rs('yellow18/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<a NAME="E5:87"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<font color="Maroon" title="c6">F2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yellow18.html#K3" title="YELLOW18:func.3">-carrier_of</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yellow18.html#K3" title="YELLOW18:func.3">-carrier_of</a> <font color="Maroon" title="c3">b</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="yellow18.html#D12" target="_self" onmouseover="rs('yellow18/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<a NAME="E6:87"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yellow18.html#K3" title="YELLOW18:func.3">-carrier_of</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="funct_2.html#T92" onmouseover="rs('funct_2/T92')" onmouseout="rh()">FUNCT_2:92</a></span>;<br/>
<a NAME="E7:87"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yellow18.html#K3" title="YELLOW18:func.3">-carrier_of</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="funct_2.html#T92" onmouseover="rs('funct_2/T92')" onmouseout="rh()">FUNCT_2:92</a></span>;<br/>
<span class="kw">consider </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">A</font>, <font color="Maroon" title="c9">f</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><br/><a NAME="E9:87"/><span class="lab"><font color="Green" title="E48">A8</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="E10:87"/><span class="lab"><font color="Green" title="E49">A9</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="E11:87"/><span class="lab"><font color="Green" title="E50">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="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="E12:87"/><span class="lab"><font color="Green" title="E51">A11</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">A</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">F1</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">f</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="yellow18.html#D12" target="_self" onmouseover="rs('yellow18/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<span class="kw">consider </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">A</font>, <font color="Maroon" title="c12">g</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><br/><a NAME="E14:87"/><span class="lab"><font color="Green" title="E52">A12</font></span>: 
<font color="Maroon" title="c10">ga</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="E15:87"/><span class="lab"><font color="Green" title="E53">A13</font></span>: 
<font color="Maroon" title="c11">gb</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="E16:87"/>
<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="E17:87"/><span class="lab"><font color="Green" title="E54">A14</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">A</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">F2</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">g</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="yellow18.html#D12" target="_self" onmouseover="rs('yellow18/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">f</font> = <font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c12">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="#E10:87"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E14:87"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E15:87"><span class="lab"><font color="Green" title="E53">A13</font></span></a></span>;<br/>
<a NAME="E19:87"/><span class="lab"><font color="Green" title="E55">A15</font></span>: 
<font color="Maroon" title="c5">F1</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"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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="c13">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p3">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</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="c2">a</font>,<font color="Maroon" title="c3">b</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="#E9:87"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="#E10:87"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E12:87"><span class="lab"><font color="Green" title="E51">A11</font></span></a></span>;<br/>
<a NAME="E20:87"/><span class="lab"><font color="Green" title="E56">A16</font></span>: 
<font color="Maroon" title="c13">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="#E10:87"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E11:87"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="altcat_1.html#D17" onmouseover="rs('altcat_1/D17')" onmouseout="rh()">ALTCAT_1:def 17</a></span>;<br/>
<a NAME="E21:87"/><span class="lab"><font color="Green" title="E57">A17</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="#E10:87"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E11:87"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="altcat_1.html#D17" onmouseover="rs('altcat_1/D17')" onmouseout="rh()">ALTCAT_1:def 17</a></span>;<br/>
<a NAME="E22:87"/>
<font color="Maroon" title="c6">F2</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"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p3">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</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="c2">a</font>,<font color="Maroon" title="c3">b</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="#E14:87"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E15:87"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="txt" href="#E17:87"><span class="lab"><font color="Green" title="E54">A14</font></span></a></span>;<br/>
<a NAME="E23:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A18</font></span>: 
<font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E19:87"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="txt" href="#E20:87"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="txt" href="#E21:87"><span class="lab"><font color="Green" title="E57">A17</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_1"><span class="kw">now </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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yellow18.html#K3" title="YELLOW18:func.3">-carrier_of</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yellow18.html#K3" title="YELLOW18:func.3">-carrier_of</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:87_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="yellow18.html#K4" title="YELLOW18:func.4">Concretized</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="yellow18.html#K3" title="YELLOW18:func.3">-carrier_of</a> <font color="Maroon" title="c2">a</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="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">bb</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c17">ff</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c16">bb</font>,<font color="Maroon" title="c2">a</font><span class="kw"> such that </span><br/><a NAME="E3:87_1"/><span class="lab"><font color="Green" title="E59">A19</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="c16">bb</font>,<font color="Maroon" title="c2">a</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="E4:87_1"/><span class="lab"><font color="Green" title="E60">A20</font></span>: 
<font color="Maroon" title="c15">x</font> <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"><font color="Maroon" title="c17">ff</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">bb</font>,<font color="Maroon" title="c2">a</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="ref" href="yellow18.html#T43" target="_self" onmouseover="rs('yellow18/T43')" onmouseout="rh()">Th43</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> = 
<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">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c17">ff</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">bb</font>,<font color="Maroon" title="c3">b</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="#E9:87"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="#E10:87"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E12:87"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E3:87_1"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E4:87_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:87"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E15:87"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="txt" href="#E17:87"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="txt" href="#E23:87"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E3:87_1"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E4:87_1"><span class="lab"><font color="Green" title="E60">A20</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="E25:87"/>
<font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E7:87"><span class="lab"><font color="Green" title="E47">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/>


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