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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="Red">::  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 <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> holds <br/> <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></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="Red">::  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 <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> holds <br/> <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></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="Red">::  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 <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> holds <br/> <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></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="Red">::  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 <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> holds <br/> <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></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="Red">::  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 <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> holds <br/> <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></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="Red">::  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> implies  <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> )</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">assume </span><span class="kw">that </span><br/><a NAME="E1:43"/><span class="lab"><font color="Green" title="E19">A1</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">and </span><br/><a NAME="E2:43"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<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">and </span><br/><a NAME="E3:43"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<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">and </span><br/><a NAME="E4:43"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<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">and </span><br/><a NAME="E5:43"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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></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/>
<div><span class="lab"><font color="Green" title="E24">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">now </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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">f</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font></span><br/><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="E2:43_1"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c10">f</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="E3:43_1"/><span class="lab"><font color="Green" title="E26">A8</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="txt" href="#E1:43"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E4:43_1"/>
 ex <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 <font color="Maroon" title="c1">C</font> st <br/>( <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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> &amp; <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:43_1"/>
<font color="Maroon" title="c10">f</font> is   <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="#E2:43_1"><span class="lab"><font color="Green" title="E25">A7</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>
<a NAME="E7:43"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">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> <font color="Maroon" title="c6">p</font> iff <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c11">g</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="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c6">p</font> iff <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> )</span><br/>

<a NAME="E1:43_2"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
( <font color="Maroon" title="c11">g</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c10">f</font> is   <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="#E6:43"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>;<br/>
<a NAME="E2:43_2"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
( <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</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="c11">g</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:43_2"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c6">p</font> implies <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> )
  <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="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c6">p</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_2_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c6">p</font>
 ; <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="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font></span><br/>

<a NAME="E2:43_2_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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> <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="txt" href="#E4:43"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E3:43_2_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:43_2_1"/>
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</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">assume </span><a NAME="E4:43_2"/>
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</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="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c6">p</font></span><br/>

<a NAME="E5:43_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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="#E1:43_2"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></span>;<br/>
<a NAME="E6:43_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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> <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="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:43_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</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="E8:43"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c11">g</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="Red">::  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="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> implies <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_3"/>
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font>
 ; <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="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>

<a NAME="E2:43_3"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c6">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E25">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_3"/>
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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="#E4:43"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</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="E9:43"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c11">g</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="Red">::  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="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> implies <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> )</span><br/>

<a NAME="E1:43_4"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
( <font color="Maroon" title="c10">f</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">g</font> is   <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="#E6:43"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:43_4"/><span class="lab"><font color="Green" title="E29">A15</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font>
 ; <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="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font></span><br/>

<a NAME="E3:43_4"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c6">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E25">A9</font></span></a></span>;<br/>
<a NAME="E4:43_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">h</font> = <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> 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="#E6:43"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>;<br/>
<a NAME="E6:43_4"/><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<font color="Maroon" title="c12">h</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="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E29">A15</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">a</font> =  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c12">h</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">b</font> =  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c12">h</font>;<br/>
<a NAME="E7:43_4"/><span class="lab"><font color="Green" title="E31">A17</font></span>: 
( <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</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="c11">g</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E8:43_4"/>
<font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</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="c11">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E9:43_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A18</font></span>: 
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">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E6:43_4"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E7:43_4"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></span>;<br/>
<a NAME="E10:43_4"/>
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E11:43_4"/><span class="kw">then </span>
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">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E6:43_4"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E7:43_4"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></span>;<br/>
<a NAME="E12:43_4"/><span class="kw">then </span>
( <font color="Maroon" title="c12">h</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">h</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">h</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">h</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">h</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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_4"><span class="lab"><font color="Green" title="E32">A18</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:43_4"/>
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:43"/><span class="lab"><font color="Green" title="E28">A19</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be   <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> #); <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">a9</font>, <font color="Olive" title="b2">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="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a9</font> &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b9</font> holds <br/> <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">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="b1">a9</font>,<font color="Olive" title="b2">b9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c12">a9</font>, <font color="Maroon" title="c13">b9</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <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="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a9</font> &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b9</font> implies  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">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><br/>



<span class="kw">assume </span><a NAME="E1:43_5"/><span class="lab"><font color="Green" title="E29">A20</font></span>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a9</font> &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b9</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">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><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">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="c14">x</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> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> implies <font color="Maroon" title="c14">x</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> <font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c13">b9</font> ) &amp; ( <font color="Maroon" title="c14">x</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> <font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c13">b9</font> implies <font color="Maroon" title="c14">x</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> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:43_5_1"/>
( <font color="Maroon" title="c14">x</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> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> implies <font color="Maroon" title="c14">x</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> <font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c13">b9</font> )
  <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="c14">x</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> <font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c13">b9</font> implies <font color="Maroon" title="c14">x</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> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_5_1_1"/><span class="lab"><font color="Green" title="E30">A21</font></span>: 
<font color="Maroon" title="c14">x</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> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font>
 ; <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="c14">x</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> <font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c13">b9</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">f</font> = <font color="Maroon" title="c14">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="c16">f9</font> = <font color="Maroon" title="c15">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="#E6:43"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>;<br/>
<a NAME="E4:43_5_1_1"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c15">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="c16">f9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E5:43_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A22</font></span>: 
<font color="Maroon" title="c11">b</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">f9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5_1_1"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<a NAME="E6:43_5_1_1"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c15">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="c16">f9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E7:43_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">a</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="c16">f9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5_1_1"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:43_5_1_1"/>
<font color="Maroon" title="c14">x</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> <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="#E1:43_5"><span class="lab"><font color="Green" title="E29">A20</font></span></a>, <a class="txt" href="#E5:43_5_1_1"><span class="lab"><font color="Green" title="E31">A22</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">assume </span><a NAME="E2:43_5_1"/><span class="lab"><font color="Green" title="E30">A23</font></span>: 
<font color="Maroon" title="c14">x</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> <font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c13">b9</font>
 ; <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="c14">x</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> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">f9</font> = <font color="Maroon" title="c14">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> ;<br/><a NAME="E4:43_5_1"/>
 <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> <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="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E29">A20</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">f</font> = <font color="Maroon" title="c15">f9</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> #) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:43_5_1"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E6:43_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="c15">f9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><a NAME="E7:43_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A24</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> <font color="Maroon" title="c13">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_5_1"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/><a NAME="E8:43_5_1"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c16">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="c15">f9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><a NAME="E9:43_5_1"/><span class="kw">then </span>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_5_1"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/><span class="kw">hence </span><a NAME="E10:43_5_1"/>
<font color="Maroon" title="c14">x</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> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E29">A20</font></span></a>, <a class="txt" href="#E7:43_5_1"><span class="lab"><font color="Green" title="E31">A24</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="E3:43_5"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">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="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</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">thus </span><a NAME="E11:43"/>
 <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="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">thus </span><a NAME="E1:43_6_1"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">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> <font color="Maroon" title="c6">p</font> iff <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E25">A9</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"> ( ( for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/>( <font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> ) ) &amp; ( for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> &amp; <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font></span>)</span>,<font color="Olive" title="b1">f</font> ) &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">O</font> holds <br/> ( <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; ( for <font color="Olive" title="b2">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; ( for <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) ) ) )</span><br/><span class="kw">thus </span><a NAME="E2:43_6_1"/><span class="lab"><font color="Green" title="E29">A25</font></span>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/>( <font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
  <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">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> &amp; <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font></span>)</span>,<font color="Olive" title="b1">f</font> ) &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">O</font> holds <br/> ( <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; ( for <font color="Olive" title="b2">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; ( for <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c11">g</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="Red">::  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="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> implies ( <font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> ) )</span><br/>

<a NAME="E1:43_6_1_1"/><span class="lab"><font color="Green" title="E30">A26</font></span>: 
( <font color="Maroon" title="c11">g</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c10">f</font> is   <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="#E6:43"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>;<br/>
<a NAME="E2:43_6_1_1"/><span class="lab"><font color="Green" title="E31">A27</font></span>: 
( <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</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="c11">g</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:43_6_1_1"/><span class="lab"><font color="Green" title="E32">A28</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font>
 ; <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="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> )</span><br/>

<a NAME="E4:43_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A29</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E27">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> = 
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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
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> <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="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E3:43_6_1_1"><span class="lab"><font color="Green" title="E32">A28</font></span></a></span>
<br/>.= 
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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_6_1_1"><span class="lab"><font color="Green" title="E32">A28</font></span></a>, <a class="txt" href="#E1:43_6_1_1"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="txt" href="#E2:43_6_1_1"><span class="lab"><font color="Green" title="E31">A27</font></span></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
;<br/> <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="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> = 
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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E4:43_6_1_1"><span class="lab"><font color="Green" title="E33">A29</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
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> <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="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E3:43_6_1_1"><span class="lab"><font color="Green" title="E32">A28</font></span></a></span>
<br/>.= 
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="c11">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_6_1_1"><span class="lab"><font color="Green" title="E32">A28</font></span></a>, <a class="txt" href="#E1:43_6_1_1"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="txt" href="#E2:43_6_1_1"><span class="lab"><font color="Green" title="E31">A27</font></span></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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">thus </span><a NAME="E3:43_6_1"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> &amp; <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font></span>)</span>,<font color="Olive" title="b1">f</font>
  <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">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">O</font> holds <br/> ( <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; ( for <font color="Olive" title="b2">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; ( for <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c11">g</font>, <font color="Maroon" title="c12">h</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="Red">::  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="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> implies <font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c11">g</font></span>)</span>,<font color="Maroon" title="c10">f</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_6_1_2"/><span class="lab"><font color="Green" title="E30">A30</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font>
 <span class="kw">and </span><br/><a NAME="E2:43_6_1_2"/><span class="lab"><font color="Green" title="E31">A31</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font>
 ; <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="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c11">g</font></span>)</span>,<font color="Maroon" title="c10">f</font></span><br/>

<a NAME="E3:43_6_1_2"/><span class="lab"><font color="Green" title="E32">A32</font></span>: 
( <font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c11">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c11">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E2:43_6_1"><span class="lab"><font color="Green" title="E29">A25</font></span></a>, <a class="txt" href="#E1:43_6_1_2"><span class="lab"><font color="Green" title="E30">A30</font></span></a></span>;<br/>
<a NAME="E4:43_6_1_2"/><span class="lab"><font color="Green" title="E33">A33</font></span>: 
( <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</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="c11">g</font> &amp; <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</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="c11">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E5:43_6_1_2"/><span class="lab"><font color="Green" title="E34">A34</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E6:43_6_1_2"/><span class="lab"><font color="Green" title="E35">A35</font></span>: 
( <font color="Maroon" title="c10">f</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">h</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="c12">h</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E7:43_6_1_2"/><span class="lab"><font color="Green" title="E36">A36</font></span>: 
( <font color="Maroon" title="c12">h</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c11">g</font> is   <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="#E6:43"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>;<br/>
<a NAME="E8:43_6_1_2"/>
( <font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E2:43_6_1"><span class="lab"><font color="Green" title="E29">A25</font></span></a>, <a class="txt" href="#E2:43_6_1_2"><span class="lab"><font color="Green" title="E31">A31</font></span></a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> = 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></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="#E8:43"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E1:43_6_1_2"><span class="lab"><font color="Green" title="E30">A30</font></span></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="c12">h</font>,<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="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E2:43_6_1_2"><span class="lab"><font color="Green" title="E31">A31</font></span></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> <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="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c11">g</font></span>)</span>,<font color="Maroon" title="c10">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6_1_2"><span class="lab"><font color="Green" title="E30">A30</font></span></a>, <a class="txt" href="#E2:43_6_1_2"><span class="lab"><font color="Green" title="E31">A31</font></span></a>, <a class="txt" href="#E7:43_6_1_2"><span class="lab"><font color="Green" title="E36">A36</font></span></a>, <a class="txt" href="#E6:43_6_1_2"><span class="lab"><font color="Green" title="E35">A35</font></span></a>, <a class="txt" href="#E4:43_6_1_2"><span class="lab"><font color="Green" title="E33">A33</font></span></a>, <a class="txt" href="#E5:43_6_1_2"><span class="lab"><font color="Green" title="E34">A34</font></span></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c11">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<font color="Maroon" title="c10">f</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="#E8:43"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E1:43_6_1_2"><span class="lab"><font color="Green" title="E30">A30</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c11">g</font></span>)</span>,<font color="Maroon" title="c10">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E2:43_6_1_2"><span class="lab"><font color="Green" title="E31">A31</font></span></a>, <a class="txt" href="#E3:43_6_1_2"><span class="lab"><font color="Green" title="E32">A32</font></span></a></span>
;<br/> <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">let </span><font color="Maroon" title="c10">b</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="Red">::  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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> &amp; ( for <font color="Olive" title="b1">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> ) &amp; ( for <font color="Olive" title="b1">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )</span><br/><a NAME="E4:43_6_1"/><span class="lab"><font color="Green" title="E30">A37</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><a NAME="E5:43_6_1"/>
( <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><span class="kw">hence </span><a NAME="E6:43_6_1"/><span class="lab"><font color="Green" title="E31">A38</font></span>: 
( <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_6_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 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">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> ) &amp; ( for <font color="Olive" title="b1">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )</span><br/><span class="kw">thus </span><a NAME="E7:43_6_1"/>
for <font color="Olive" title="b1">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>
  <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">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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="Red">::  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="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> implies <font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_6_1_3"/><span class="lab"><font color="Green" title="E32">A39</font></span>: 
<font color="Maroon" title="c5">c</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> <font color="Maroon" title="c10">b</font>
 ; <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="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font></span><br/>

<a NAME="E2:43_6_1_3"/><span class="lab"><font color="Green" title="E33">A40</font></span>: 
( <font color="Maroon" title="c11">f</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c5">c</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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<font color="Maroon" title="c11">f</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="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<font color="Maroon" title="c11">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E6:43_6_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:43_6_1_3"><span class="lab"><font color="Green" title="E32">A39</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_6_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E1:43_6_1_3"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E2:43_6_1_3"><span class="lab"><font color="Green" title="E33">A40</font></span></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</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">let </span><font color="Maroon" title="c11">g</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="Red">::  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="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> implies <font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> )</span><br/><a NAME="E8:43_6_1"/><span class="lab"><font color="Green" title="E32">A41</font></span>: 
( <font color="Maroon" title="c11">g</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</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="c11">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><span class="kw">assume </span><a NAME="E9:43_6_1"/><span class="lab"><font color="Green" title="E33">A42</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 ; <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="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font></span><br/><span class="kw">hence </span><font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
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="c11">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E6:43_6_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_6_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E9:43_6_1"><span class="lab"><font color="Green" title="E33">A42</font></span></a>, <a class="txt" href="#E8:43_6_1"><span class="lab"><font color="Green" title="E32">A41</font></span></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></span>
;<br/> <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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">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> #) as   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></span>;<br/>
<a NAME="E3:43_6"/><span class="lab"><font color="Green" title="E29">A43</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="c10">B</font><br/> for <font color="Olive" title="b2">a9</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="b2">a9</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>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive" title="b2">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E4:43_6"/>
( ( 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 <font color="Maroon" title="c10">B</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="tarski.html#R1" title="TARSKI:pred.1">c=</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> ) &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c10">B</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="#E4:43"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:43_6"/>
 <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">by</span> <span class="lab"><a class="txt" href="#E3:43_6"><span class="lab"><font color="Green" title="E29">A43</font></span></a>, <a class="ref" href="cat_2.html#D4" target="_self" title="CAT_2:def.4">Def4</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">thus </span><a NAME="E12:43"/>
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="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E28">A19</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>
