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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">O</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">M</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b3">d</font>, <font color="Olive" title="b4">c</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font><br/>  for <font color="Olive" title="b5">p</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b2">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Olive" title="b2">M</font><br/>  for <font color="Olive" title="b6">i</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">M</font>  st  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Olive" title="b1">O</font>,<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">d</font>,<font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">i</font> #) <a href="cat_2.html#R1" title="CAT_2:pred.1">is_full_subcategory_of</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b7">a</font>,<font color="Olive" title="b8">b</font>)</span>)</span> where <font color="Olive" title="b7">a</font>, <font color="Olive" title="b8">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp; <font color="Olive" title="b8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> ) </span> } </span>  &amp; <font color="Olive" title="b3">d</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b4">c</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b5">p</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="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b6">i</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">O</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">O</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">d</font>, <font color="Olive" title="b3">c</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c2">O</font><br/>  for <font color="Olive" title="b4">p</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Olive" title="b1">M</font><br/>  for <font color="Olive" title="b5">i</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">O</font>,<font color="Olive" title="b1">M</font>  st  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">d</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">i</font> #) <a href="cat_2.html#R1" title="CAT_2:pred.1">is_full_subcategory_of</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b6">a</font>,<font color="Olive" title="b7">b</font>)</span>)</span> where <font color="Olive" title="b6">a</font>, <font color="Olive" title="b7">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span>  &amp; <font color="Olive" title="b2">d</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">c</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b4">p</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="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b5">i</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">d</font>, <font color="Olive" title="b2">c</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">O</font><br/>  for <font color="Olive" title="b3">p</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c3">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c3">M</font><br/>  for <font color="Olive" title="b4">i</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>  st  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">c</font>,<font color="Olive" title="b3">p</font>,<font color="Olive" title="b4">i</font> #) <a href="cat_2.html#R1" title="CAT_2:pred.1">is_full_subcategory_of</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font>)</span>)</span> where <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span>  &amp; <font color="Olive" title="b1">d</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b2">c</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b3">p</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="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b4">i</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">d</font>, <font color="Maroon" title="c5">c</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c3">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c3">M</font><br/>  for <font color="Olive" title="b2">i</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>  st  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">i</font> #) <a href="cat_2.html#R1" title="CAT_2:pred.1">is_full_subcategory_of</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>)</span>)</span> where <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span>  &amp; <font color="Maroon" title="c4">d</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">c</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b1">p</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="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b2">i</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">p</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c3">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>  st  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Olive" title="b1">i</font> #) <a href="cat_2.html#R1" title="CAT_2:pred.1">is_full_subcategory_of</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>)</span>)</span> where <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span>  &amp; <font color="Maroon" title="c4">d</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">c</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c6">p</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="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b1">i</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">i</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) <a href="cat_2.html#R1" title="CAT_2:pred.1">is_full_subcategory_of</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span>  &amp; <font color="Maroon" title="c4">d</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">c</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c6">p</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="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c7">i</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font> ) )</span><br/>











<span class="kw">set </span><font color="Maroon" title="c8">H</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">B</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #);<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E2:35"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #)<br/>  for <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a9</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b9</font> holds <br/> <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b3">a9</font>,<font color="Olive" title="b4">b9</font>)
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_2.html#D6" target="_self" title="CAT_2:def.6">CAT_2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span>  &amp; <font color="Maroon" title="c4">d</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">c</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c6">p</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="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c7">i</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font> )</span><br/>

<a NAME="E3:35"/><span class="lab"><font color="Green" title="E21">A3</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 <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) holds <br/> ( <font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</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="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</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="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</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="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</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="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">f9</font> = <font color="Maroon" title="c10">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="cat_2.html#T8" target="_self" title="CAT_2:th.8">Th14</a></span>;<br/>
<a NAME="E2:35_1"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">f9</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">f</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="Maroon" title="c11">f9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="cat_2.html#T9" target="_self" title="CAT_2:th.9">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:35_1"/>
( <font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</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="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</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="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> )
 ; <a class="txt" 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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span>  ) &amp; ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span>  implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:35_2"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span>  implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c10">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">f9</font> = <font color="Maroon" title="c11">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="cat_2.html#T8" target="_self" title="CAT_2:th.8">Th14</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">a9</font> =  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c12">f9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">b9</font> =  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c12">f9</font>;<br/>
<a NAME="E4:35_2_1"/>
(  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">f</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="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f9</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">f</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="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>;<br/>
<a NAME="E5:35_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c12">f9</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c12">f9</font></span>)</span>) &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c12">f9</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c12">f9</font></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E6:35_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" 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">assume </span><a NAME="E2:35_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:35_2"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 <span class="kw">and </span><br/><a NAME="E5:35_2"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">a9</font>, <font color="Maroon" title="c13">b9</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E7:35_2"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c13">b9</font>)
 <span class="kw">and </span><br/><a NAME="E8:35_2"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
( <font color="Maroon" title="c12">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Maroon" title="c13">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_2"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c12">a9</font>, <font color="Maroon" title="c15">b</font> = <font color="Maroon" title="c13">b9</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_2"><span class="lab"><font color="Green" title="E25">A7</font></span></a></span>;<br/><a NAME="E10:35_2"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c13">b9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E20">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:35_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_2"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E7:35_2"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>; <a class="txt" 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="E5:35"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">d</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">c</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c6">p</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="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c7">i</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">d9</font> =  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c11">c9</font> =  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">O</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T20" target="_self" title="CAT_2:th.20">Th29</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">p9</font> =  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font></span><br/><a NAME="E1:35_3"/>
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</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="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:35_3"/>
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>; <a class="txt" 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="E8:35"/>
<font color="Maroon" title="c4">d</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">c</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c6">p</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="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c7">i</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font></span><br/><a NAME="E1:35_4"/>
<font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</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="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:35_4"/>
<font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>; <a class="txt" 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="E10:35"/>
<font color="Maroon" title="c5">c</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">p</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="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c7">i</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_5"><span class="kw">now </span></a><div class="add"><a NAME="E1:35_5"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">p</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"><font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c3">M</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/><a NAME="E2:35_5"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="c1">C</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c3">M</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#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/><a NAME="E3:35_5"/>
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="cat_2.html#D4" target="_self" title="CAT_2:def.4">Def4</a></span>;<br/><a NAME="E4:35_5"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/><a NAME="E5:35_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A11</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_5"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E2:35_5"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/><a NAME="E6:35_5"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">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> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span> or <font color="Maroon" title="c13">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> <font color="Maroon" title="c6">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_5_1"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
<font color="Maroon" title="c13">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> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">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> <font color="Maroon" title="c6">p</font></span><br/>

<a NAME="E2:35_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c3">M</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="#E2:35_5"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">g</font>, <font color="Maroon" title="c15">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font><span class="kw"> such that </span><br/><a NAME="E4:35_5_1"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T1" title="DOMAIN_1:th.1">DOMAIN_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">f</font> = <font color="Maroon" title="c15">f</font>, <font color="Maroon" title="c17">g</font> = <font color="Maroon" title="c14">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">f9</font> = <font color="Maroon" title="c16">f</font>, <font color="Maroon" title="c19">g9</font> = <font color="Maroon" title="c17">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="cat_2.html#T8" target="_self" title="CAT_2:th.8">Th14</a></span>;<br/>
<a NAME="E7:35_5_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">g</font>,<font color="Maroon" title="c16">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_5"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E1:35_5_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E4:35_5_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E8:35_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A14</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c19">g9</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="Maroon" title="c18">f9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T15" title="CAT_1:th.15">CAT_1:15</a></span>;<br/>
<a NAME="E9:35_5_1"/>
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c16">f</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="Maroon" title="c18">f9</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c19">g9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="cat_2.html#T9" target="_self" title="CAT_2:th.9">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:35_5_1"/>
<font color="Maroon" title="c13">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> <font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E4:35_5_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E8:35_5_1"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="cat_1.html#T15" title="CAT_1:th.15">CAT_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:35_5"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_5"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" 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">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> <font color="Maroon" title="c6">p</font> holds <br/><font color="Maroon" title="c6">p</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> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span> <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="c13">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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> <font color="Maroon" title="c6">p</font> implies <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/><span class="kw">assume </span><a NAME="E8:35_5"/><span class="lab"><font color="Green" title="E26">A15</font></span>: 
<font color="Maroon" title="c13">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> <font color="Maroon" title="c6">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">g</font>, <font color="Maroon" title="c15">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font><span class="kw"> such that </span><br/><a NAME="E10:35_5"/><span class="lab"><font color="Green" title="E27">A16</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_5"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="domain_1.html#T1" title="DOMAIN_1:th.1">DOMAIN_1:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">f</font> = <font color="Maroon" title="c15">f</font>, <font color="Maroon" title="c17">g</font> = <font color="Maroon" title="c14">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) ;<br/><a NAME="E12:35_5"/><span class="lab"><font color="Green" title="E28">A17</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c17">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="Maroon" title="c16">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E8:35_5"><span class="lab"><font color="Green" title="E26">A15</font></span></a>, <a class="txt" href="#E10:35_5"><span class="lab"><font color="Green" title="E27">A16</font></span></a>, <a class="ref" href="cat_1.html#T15" title="CAT_1:th.15">CAT_1:15</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">f9</font> = <font color="Maroon" title="c16">f</font>, <font color="Maroon" title="c19">g9</font> = <font color="Maroon" title="c17">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="cat_2.html#T8" target="_self" title="CAT_2:th.8">Th14</a></span>;<br/><a NAME="E14:35_5"/><span class="lab"><font color="Green" title="E29">A18</font></span>: 
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c16">f</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="Maroon" title="c18">f9</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c19">g9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="cat_2.html#T9" target="_self" title="CAT_2:th.9">Th15</a></span>;<br/><a NAME="E15:35_5"/>
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">g</font>,<font color="Maroon" title="c16">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_5"><span class="lab"><font color="Green" title="E27">A16</font></span></a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> = 
<font color="Maroon" title="c17">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c16">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E12:35_5"><span class="lab"><font color="Green" title="E28">A17</font></span></a>, <a class="ref" href="cat_1.html#T16" title="CAT_1:th.16">CAT_1:16</a></span>
<br/>.= 
<font color="Maroon" title="c19">g9</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c18">f9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E12:35_5"><span class="lab"><font color="Green" title="E28">A17</font></span></a>, <a class="ref" href="cat_2.html#T11" target="_self" title="CAT_2:th.11">Th17</a></span>
<br/>.= 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c19">g9</font>,<font color="Maroon" title="c18">f9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_5"><span class="lab"><font color="Green" title="E29">A18</font></span></a>, <a class="txt" href="#E12:35_5"><span class="lab"><font color="Green" title="E28">A17</font></span></a>, <a class="ref" href="cat_1.html#T16" title="CAT_1:th.16">CAT_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_5"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E8:35_5"><span class="lab"><font color="Green" title="E26">A15</font></span></a>, <a class="txt" href="#E10:35_5"><span class="lab"><font color="Green" title="E27">A16</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
;<br/> <a class="txt" 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="E12:35"/>
<font color="Maroon" title="c6">p</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="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">i</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">i9</font> =  the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="ref" href="cat_2.html#T20" target="_self" title="CAT_2:th.20">Th29</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c15">b</font> = <font color="Maroon" title="c14">a</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">a9</font> = <font color="Maroon" title="c14">a</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ;<br/><a NAME="E3:35_6_1"/>
 <a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c16">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="cat_2.html#D4" target="_self" title="CAT_2:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_6_1"/>
<font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">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="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Maroon" title="c14">a</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#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font> )</span><br/><span class="kw">thus </span><a NAME="E5:35_6_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</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#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font></span><br/><a NAME="E6:35_6_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 ;<br/><span class="kw">hence </span><a NAME="E7:35_6_1"/>
<font color="Maroon" title="c14">a</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#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>; <a class="txt" 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="E2:35_6"/>
<font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>; <a class="txt" 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="E15:35"/>
<font color="Maroon" title="c7">i</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="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>; <a class="txt" 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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