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<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span>,<span class="p3">(<span class="default">$1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">Ob3</font> = <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">G</font> =  <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">o</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:26_1_1"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:26_1_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> holds <br/><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">e</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c5">o</font> = <font color="Maroon" title="c4">o</font> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">o</font> implies <font color="Maroon" title="c5">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )</span><br/><span class="kw">assume </span><a NAME="E1:26_1_1_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/><a NAME="E2:26_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E3:26_1_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_1_1_1"/>
<font color="Maroon" title="c5">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">o</font> = <font color="Maroon" title="c5">o</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> implies <font color="Maroon" title="c6">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c6">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><span class="kw">assume </span><a NAME="E2:26_1_1_2"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font>, <font color="Maroon" title="c11">k</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E4:26_1_1_2"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T3" title="DOMAIN_1:th.3">DOMAIN_1:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">e9</font> = <font color="Maroon" title="c7">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1_2"><span class="lab"><font color="Green" title="E11">A3</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E13">A5</font></span>: <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  = 
<font color="Maroon" title="c12">e9</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="mcart_1.html#T46" title="MCART_1:th.46">MCART_1:46</a></span>
<br/>.= 
<font color="Maroon" title="c10">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="mcart_1.html#T43" title="MCART_1:th.43">MCART_1:43</a></span>
;<br/><span class="lab"><font color="Green" title="E14">A6</font></span>: <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  = 
<font color="Maroon" title="c12">e9</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="mcart_1.html#T46" title="MCART_1:th.46">MCART_1:46</a></span>
<br/>.= 
<font color="Maroon" title="c11">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="mcart_1.html#T43" title="MCART_1:th.43">MCART_1:43</a></span>
;<br/><span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  = 
<font color="Maroon" title="c12">e9</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="mcart_1.html#T46" title="MCART_1:th.46">MCART_1:46</a></span>
<br/>.= 
<font color="Maroon" title="c9">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="mcart_1.html#T43" title="MCART_1:th.43">MCART_1:43</a></span>
;<br/><a NAME="E9:26_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c8">f</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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>)</span>)</span></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="#E3:26_1_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:26_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E6:26_1_1_2"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E7:26_1_1_2"><span class="lab"><font color="Green" title="E14">A6</font></span></a></span>;<br/><a NAME="E10:26_1_1_2"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
( <span class="p1">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</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="c9">i</font>,<font color="Maroon" title="c10">j</font>) &amp; <span class="p1">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</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="c10">j</font>,<font color="Maroon" title="c11">k</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_2.html#D3" target="_self" title="ALTCAT_2:def.3">Def3</a></span>;<br/><span class="lab"><font color="Green" title="E17">A9</font></span>: <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> = 
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">k</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D3" title="ALTCAT_1:def.3">ALTCAT_1:def 3</a></span>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">k</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_2.html#D3" target="_self" title="ALTCAT_2:def.3">Def3</a></span>
;<br/><span class="lab"><font color="Green" title="E18">A10</font></span>: <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> = 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></span>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>)</span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_1_1_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></span>
;<br/><div><span class="lab"><font color="Green" title="E19">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2_6"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_1_1_2_6"/>
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><a NAME="E2:26_1_1_2_6"/><span class="kw">then </span>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_1_1_2"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="cat_1.html#T24" title="CAT_1:th.24">CAT_1:24</a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_1_1_2_6"/>
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_1_1_2"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T90" title="ZFMISC_1:th.90">ZFMISC_1:90</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E14:26_1_1_2"/>
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>)</span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <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="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_2.html#T11" target="_self" title="ALTCAT_2:th.11">Th12</a></span>;<br/><a NAME="E15:26_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A12</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_1_1_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:26_1_1_2"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E11:26_1_1_2"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="relat_1.html#T115" title="RELAT_1:th.115">RELAT_1:115</a></span>;<br/><a NAME="E16:26_1_1_2"/>
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>)</span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  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="altcat_2.html#T10" target="_self" title="ALTCAT_2:th.10">Th11</a></span>;<br/><a NAME="E17:26_1_1_2"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_1_1_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:26_1_1_2"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E12:26_1_1_2"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/><span class="kw">hence </span><a NAME="E18:26_1_1_2"/>
<font color="Maroon" title="c6">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26_1_1_2"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E15:26_1_1_2"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">o</font> = <font color="Maroon" title="c6">o</font> as   <a href="altcat_1.html#NM3" title="ALTCAT_1:NM.3">BinComp</a> of <span class="p1">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D15" title="PBOOLE:def.15">PBOOLE:def 15</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">o</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">k</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c7">o</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">j</font>, <font color="Maroon" title="c10">k</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">o</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">e</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> ;<br/>
<span class="lab"><font color="Green" title="E11">A13</font></span>: <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  = 
<font color="Maroon" title="c11">e</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T46" title="MCART_1:th.46">MCART_1:46</a></span>
<br/>.= 
<font color="Maroon" title="c8">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T43" title="MCART_1:th.43">MCART_1:43</a></span>
;<br/>
<span class="lab"><font color="Green" title="E12">A14</font></span>: <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  = 
<font color="Maroon" title="c11">e</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T46" title="MCART_1:th.46">MCART_1:46</a></span>
<br/>.= 
<font color="Maroon" title="c9">j</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T43" title="MCART_1:th.43">MCART_1:43</a></span>
;<br/>
<span class="lab"><font color="Green" title="E13">A15</font></span>: <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  = 
<font color="Maroon" title="c11">e</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T46" title="MCART_1:th.46">MCART_1:46</a></span>
<br/>.= 
<font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T43" title="MCART_1:th.43">MCART_1:43</a></span>
;<br/>
<span class="kw">thus </span><font color="Maroon" title="c7">o</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font>) = 
<font color="Maroon" title="c7">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K2" title="ALTCAT_2:func.2">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>)</span>)</span></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="#E3:26_1_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E10:26_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E11:26_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E12:26_1_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
