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<span class="kw">let </span><font color="Maroon" title="c3">C1</font>, <font color="Maroon" title="c4">C2</font> be   <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a>  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; ( for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font> holds <br/> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  ) ) &amp; ( for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> ) &amp; ( for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font>  st <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> holds <br/> ex <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b6">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font> ex <font color="Olive" title="b7">t1</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> st <br/>( <font color="Olive" title="b1">f</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b6">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) ) &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C1</font><br/> for <font color="Olive" title="b2">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive" title="b1">a</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F</font>,<font color="Olive" title="b2">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Olive" title="b2">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; ( for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font> holds <br/> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  ) ) &amp; ( for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> ) &amp; ( for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font>  st <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> holds <br/> ex <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b6">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font> ex <font color="Olive" title="b7">t1</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> st <br/>( <font color="Olive" title="b1">f</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b6">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) ) &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c4">C2</font><br/> for <font color="Olive" title="b2">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive" title="b1">a</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F</font>,<font color="Olive" title="b2">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Olive" title="b2">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) implies <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_1_2"/><span class="lab"><font color="Green" title="E47">A103</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E2:61_1_2"/><span class="lab"><font color="Green" title="E48">A104</font></span>: 
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E3:61_1_2"/><span class="lab"><font color="Green" title="E49">A105</font></span>: 
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font> holds <br/> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 <span class="kw">and </span><br/><a NAME="E4:61_1_2"/><span class="lab"><font color="Green" title="E50">A106</font></span>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font>
 <span class="kw">and </span><br/><a NAME="E5:61_1_2"/><span class="lab"><font color="Green" title="E51">A107</font></span>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font>  st <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> holds <br/> ex <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b6">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font> ex <font color="Olive" title="b7">t1</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> st <br/>( <font color="Olive" title="b1">f</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b6">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">and </span><br/><a NAME="E6:61_1_2"/><span class="lab"><font color="Green" title="E52">A108</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C1</font><br/> for <font color="Olive" title="b2">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive" title="b1">a</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F</font>,<font color="Olive" title="b2">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Olive" title="b2">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:61_1_2"/><span class="lab"><font color="Green" title="E53">A109</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E8:61_1_2"/><span class="lab"><font color="Green" title="E54">A110</font></span>: 
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E9:61_1_2"/><span class="lab"><font color="Green" title="E55">A111</font></span>: 
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font> holds <br/> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 <span class="kw">and </span><br/><a NAME="E10:61_1_2"/><span class="lab"><font color="Green" title="E56">A112</font></span>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font>
 <span class="kw">and </span><br/><a NAME="E11:61_1_2"/><span class="lab"><font color="Green" title="E57">A113</font></span>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font>  st <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> holds <br/> ex <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b6">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font> ex <font color="Olive" title="b7">t1</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> st <br/>( <font color="Olive" title="b1">f</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b6">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">and </span><br/><a NAME="E12:61_1_2"/><span class="lab"><font color="Green" title="E58">A114</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c4">C2</font><br/> for <font color="Olive" title="b2">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive" title="b1">a</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F</font>,<font color="Olive" title="b2">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Olive" title="b2">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font></span><br/>

<a NAME="E13:61_1_2"/><span class="lab"><font color="Green" title="E59">A115</font></span>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">C2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:61_1_2_1"/>
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E48">A104</font></span></a>, <a class="txt" href="#E8:61_1_2"><span class="lab"><font color="Green" title="E54">A110</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_2.html#D8" title="FUNCT_2:def.8">FUNCT_2:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C1</font> holds  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c5">a</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E48">A104</font></span></a>, <a class="txt" href="#E8:61_1_2"><span class="lab"><font color="Green" title="E54">A110</font></span></a></span>;<br/>
<span class="kw">thus </span>the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> = 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">a</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2"><span class="lab"><font color="Green" title="E49">A105</font></span></a></span>
<br/>.= 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_2"><span class="lab"><font color="Green" title="E55">A111</font></span></a></span>
<br/>.= 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E60">A116</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">S1</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font>, <font color="Maroon" title="c6">S2</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C1</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E48">A104</font></span></a>, <a class="txt" href="#E8:61_1_2"><span class="lab"><font color="Green" title="E54">A110</font></span></a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C1</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font> iff <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font> )</span><br/><a NAME="E1:61_1_2_2_1"/><span class="lab"><font color="Green" title="E61">A117</font></span>: 
<font color="Maroon" title="c7">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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</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="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> , <font color="Maroon" title="c9">g1</font> = <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">f2</font> = <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> , <font color="Maroon" title="c11">g2</font> = <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E48">A104</font></span></a>, <a class="txt" href="#E8:61_1_2"><span class="lab"><font color="Green" title="E54">A110</font></span></a>, <a class="txt" href="#E1:61_1_2_2_1"><span class="lab"><font color="Green" title="E61">A117</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><div><span class="lab"><font color="Green" title="E62">A118</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:61_1_2_2_1_1"/>
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c11">g2</font>,<font color="Maroon" title="c10">f2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">g1</font>,<font color="Maroon" title="c8">f1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">F</font>, <font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c14">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c15">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c12">F</font>,<font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c16">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font><span class="kw"> such that </span><br/><a NAME="E3:61_1_2_2_1_1"/><span class="lab"><font color="Green" title="E63">A119</font></span>: 
<font color="Maroon" title="c10">f2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F</font>,<font color="Maroon" title="c13">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:61_1_2_2_1_1"/><span class="lab"><font color="Green" title="E64">A120</font></span>: 
<font color="Maroon" title="c11">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:61_1_2_2_1_1"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c11">g2</font>,<font color="Maroon" title="c10">f2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c15">t</font></span>)</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="#E11:61_1_2"><span class="lab"><font color="Green" title="E57">A113</font></span></a></span>;<br/> <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c9">g1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2"><span class="lab"><font color="Green" title="E49">A105</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_2_2_1_1"><span class="lab"><font color="Green" title="E64">A120</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c13">F1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F</font>,<font color="Maroon" title="c13">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_2_1_1"><span class="lab"><font color="Green" title="E63">A119</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c8">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2"><span class="lab"><font color="Green" title="E49">A105</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E7:61_1_2_2_1_1"/>
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">g1</font>,<font color="Maroon" title="c8">f1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_2"><span class="lab"><font color="Green" title="E50">A106</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:61_1_2_2_1_2"/>
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">g1</font>,<font color="Maroon" title="c8">f1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c11">g2</font>,<font color="Maroon" title="c10">f2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">F</font>, <font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c14">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c15">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c12">F</font>,<font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c16">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font><span class="kw"> such that </span><br/><a NAME="E3:61_1_2_2_1_2"/><span class="lab"><font color="Green" title="E63">A121</font></span>: 
<font color="Maroon" title="c8">f1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F</font>,<font color="Maroon" title="c13">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:61_1_2_2_1_2"/><span class="lab"><font color="Green" title="E64">A122</font></span>: 
<font color="Maroon" title="c9">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:61_1_2_2_1_2"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">g1</font>,<font color="Maroon" title="c8">f1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c15">t</font></span>)</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="#E5:61_1_2"><span class="lab"><font color="Green" title="E51">A107</font></span></a></span>;<br/> <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">g2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_2"><span class="lab"><font color="Green" title="E55">A111</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_2_2_1_2"><span class="lab"><font color="Green" title="E64">A122</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c13">F1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F</font>,<font color="Maroon" title="c13">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_2_1_2"><span class="lab"><font color="Green" title="E63">A121</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">f2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_2"><span class="lab"><font color="Green" title="E55">A111</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E7:61_1_2_2_1_2"/>
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c11">g2</font>,<font color="Maroon" title="c10">f2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_2"><span class="lab"><font color="Green" title="E56">A112</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:61_1_2_2_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font> iff <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_2_2_1"><span class="lab"><font color="Green" title="E62">A118</font></span></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:61_1_2_2"/><span class="lab"><font color="Green" title="E61">A123</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T8" title="SUBSET_1:th.8">SUBSET_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> holds <br/>the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</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> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> implies the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/><span class="kw">assume </span><a NAME="E4:61_1_2_2"/><span class="lab"><font color="Green" title="E62">A124</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E5:61_1_2_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C1</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> , <font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_2_2"><span class="lab"><font color="Green" title="E62">A124</font></span></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></span>;<br/><a NAME="E7:61_1_2_2"/><span class="lab"><font color="Green" title="E63">A126</font></span>: 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_2_2"><span class="lab"><font color="Green" title="E62">A124</font></span></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">F9</font>, <font color="Maroon" title="c11">F19</font>, <font color="Maroon" title="c12">F29</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c13">t9</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">F9</font>,<font color="Maroon" title="c11">F19</font>, <font color="Maroon" title="c14">t19</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c11">F19</font>,<font color="Maroon" title="c12">F29</font><span class="kw"> such that </span><br/><a NAME="E9:61_1_2_2"/><span class="lab"><font color="Green" title="E64">A127</font></span>: 
<font color="Maroon" title="c8">f</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F9</font>,<font color="Maroon" title="c11">F19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">t9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:61_1_2_2"/><span class="lab"><font color="Green" title="E65">A128</font></span>: 
<font color="Maroon" title="c9">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F19</font>,<font color="Maroon" title="c12">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">t19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E11:61_1_2_2"/><span class="lab"><font color="Green" title="E66">A129</font></span>: 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F9</font>,<font color="Maroon" title="c12">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">t19</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c13">t9</font></span>)</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="#E2:61_1_2"><span class="lab"><font color="Green" title="E48">A104</font></span></a>, <a class="txt" href="#E8:61_1_2"><span class="lab"><font color="Green" title="E54">A110</font></span></a>, <a class="txt" href="#E11:61_1_2"><span class="lab"><font color="Green" title="E57">A113</font></span></a>, <a class="txt" href="#E3:61_1_2_2"><span class="lab"><font color="Green" title="E61">A123</font></span></a>, <a class="txt" href="#E4:61_1_2_2"><span class="lab"><font color="Green" title="E62">A124</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">F</font>, <font color="Maroon" title="c16">F1</font>, <font color="Maroon" title="c17">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c18">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c15">F</font>,<font color="Maroon" title="c16">F1</font>, <font color="Maroon" title="c19">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c16">F1</font>,<font color="Maroon" title="c17">F2</font><span class="kw"> such that </span><br/><a NAME="E13:61_1_2_2"/><span class="lab"><font color="Green" title="E67">A130</font></span>: 
<font color="Maroon" title="c8">f</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F</font>,<font color="Maroon" title="c16">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E14:61_1_2_2"/><span class="lab"><font color="Green" title="E68">A131</font></span>: 
<font color="Maroon" title="c9">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">F1</font>,<font color="Maroon" title="c17">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c19">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E15:61_1_2_2"/><span class="lab"><font color="Green" title="E69">A132</font></span>: 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F</font>,<font color="Maroon" title="c17">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c18">t</font></span>)</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="#E5:61_1_2"><span class="lab"><font color="Green" title="E51">A107</font></span></a>, <a class="txt" href="#E4:61_1_2_2"><span class="lab"><font color="Green" title="E62">A124</font></span></a>, <a class="txt" href="#E7:61_1_2_2"><span class="lab"><font color="Green" title="E63">A126</font></span></a></span>;<br/><a NAME="E16:61_1_2_2"/><span class="lab"><font color="Green" title="E70">A133</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F</font>,<font color="Maroon" title="c16">F1</font></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"><font color="Maroon" title="c10">F9</font>,<font color="Maroon" title="c11">F19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:61_1_2_2"><span class="lab"><font color="Green" title="E67">A130</font></span></a>, <a class="txt" href="#E9:61_1_2_2"><span class="lab"><font color="Green" title="E64">A127</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E17:61_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A134</font></span>: 
<font color="Maroon" title="c15">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c10">F9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E18:61_1_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">F1</font>,<font color="Maroon" title="c17">F2</font></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"><font color="Maroon" title="c11">F19</font>,<font color="Maroon" title="c12">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:61_1_2_2"><span class="lab"><font color="Green" title="E68">A131</font></span></a>, <a class="txt" href="#E10:61_1_2_2"><span class="lab"><font color="Green" title="E65">A128</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E19:61_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A135</font></span>: 
<font color="Maroon" title="c17">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">F29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E20:61_1_2_2"/><span class="lab"><font color="Green" title="E73">A136</font></span>: 
<font color="Maroon" title="c16">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">F19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_1_2_2"><span class="lab"><font color="Green" title="E70">A133</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E21:61_1_2_2"/>
<font color="Maroon" title="c18">t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">t9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:61_1_2_2"><span class="lab"><font color="Green" title="E67">A130</font></span></a>, <a class="txt" href="#E9:61_1_2_2"><span class="lab"><font color="Green" title="E64">A127</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">hence </span><a NAME="E22:61_1_2_2"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1_2_2"><span class="lab"><font color="Green" title="E63">A126</font></span></a>, <a class="txt" href="#E14:61_1_2_2"><span class="lab"><font color="Green" title="E68">A131</font></span></a>, <a class="txt" href="#E15:61_1_2_2"><span class="lab"><font color="Green" title="E69">A132</font></span></a>, <a class="txt" href="#E10:61_1_2_2"><span class="lab"><font color="Green" title="E65">A128</font></span></a>, <a class="txt" href="#E11:61_1_2_2"><span class="lab"><font color="Green" title="E66">A129</font></span></a>, <a class="txt" href="#E17:61_1_2_2"><span class="lab"><font color="Green" title="E71">A134</font></span></a>, <a class="txt" href="#E20:61_1_2_2"><span class="lab"><font color="Green" title="E73">A136</font></span></a>, <a class="txt" href="#E19:61_1_2_2"><span class="lab"><font color="Green" title="E72">A135</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E15:61_1_2"/><span class="lab"><font color="Green" title="E61">A137</font></span>: 
the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c4">C2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:61_1_2_3"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2"><span class="lab"><font color="Green" title="E47">A103</font></span></a>, <a class="txt" href="#E7:61_1_2"><span class="lab"><font color="Green" title="E53">A109</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_2.html#D8" title="FUNCT_2:def.8">FUNCT_2:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C1</font> holds  the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c5">a</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c4">C2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2"><span class="lab"><font color="Green" title="E47">A103</font></span></a>, <a class="txt" href="#E7:61_1_2"><span class="lab"><font color="Green" title="E53">A109</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c5">a</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2"><span class="lab"><font color="Green" title="E47">A103</font></span></a>, <a class="ref" href="cat_2.html#D3" title="CAT_2:def.3">CAT_2:def 3</a></span>;<br/>
<span class="kw">thus </span>the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> = 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c5">a</font>

<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c7">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c7">F</font></span>)</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="#E6:61_1_2"><span class="lab"><font color="Green" title="E52">A108</font></span></a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c6">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_2"><span class="lab"><font color="Green" title="E58">A114</font></span></a></span>
<br/>.= 
the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:61_1_2"/>
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">C2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:61_1_2_4"/>
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E48">A104</font></span></a>, <a class="txt" href="#E8:61_1_2"><span class="lab"><font color="Green" title="E54">A110</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_2.html#D8" title="FUNCT_2:def.8">FUNCT_2:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">C1</font> holds  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c5">a</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E48">A104</font></span></a>, <a class="txt" href="#E8:61_1_2"><span class="lab"><font color="Green" title="E54">A110</font></span></a></span>;<br/>
<span class="kw">thus </span>the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> = 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">a</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2"><span class="lab"><font color="Green" title="E49">A105</font></span></a></span>
<br/>.= 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_2"><span class="lab"><font color="Green" title="E55">A111</font></span></a></span>
<br/>.= 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E17:61_1_2"/>
<font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2"><span class="lab"><font color="Green" title="E47">A103</font></span></a>, <a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E48">A104</font></span></a>, <a class="txt" href="#E7:61_1_2"><span class="lab"><font color="Green" title="E53">A109</font></span></a>, <a class="txt" href="#E8:61_1_2"><span class="lab"><font color="Green" title="E54">A110</font></span></a>, <a class="txt" href="#E13:61_1_2"><span class="lab"><font color="Green" title="E59">A115</font></span></a>, <a class="txt" href="#E14:61_1_2"><span class="lab"><font color="Green" title="E60">A116</font></span></a>, <a class="txt" href="#E15:61_1_2"><span class="lab"><font color="Green" title="E61">A137</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
