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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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>  <a href="altcat_1.html#V7" title="ALTCAT_1:attr.7">compositional</a>  <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> ($2,$3)</span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> ($1,$2)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> =  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> ($1,$2)</span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> ($2,$3)</span>)</span>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">M</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><span class="kw"> such that </span><br/><a NAME="E2:37_1_1"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon" title="c2">M</font> <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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="altcat_1.html#S1" target="_self" title="ALTCAT_1:sch.1">ALTCAT_1:sch 1</a>();<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c3">c</font> being   <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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E4:37_1_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 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="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon" title="c3">c</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="altcat_1.html#S3" target="_self" title="ALTCAT_1:sch.3">ALTCAT_1:sch 3</a>();<br/></span>
<a NAME="E5:37_1_1"/>
<font color="Maroon" title="c3">c</font> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c3">c</font> or <font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_1_1_1"/>
<font color="Maroon" title="c4">i</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="c3">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="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<a NAME="E2:37_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</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="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font>, <font color="Maroon" title="c7">x3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:37_1_1_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c7">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <span class="kw">and </span><br/><a NAME="E5:37_1_1_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></span>;<br/>
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> = 
<font color="Maroon" title="c3">c</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></span>
<br/>.= 
 <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E4:37_1_1_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:37_1_1_1"/>
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is    <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c3">c</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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</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="c5">i</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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> implies <font color="Maroon" title="c4">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></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="c5">i</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"><font color="Maroon" title="c2">M</font></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="c5">i</font></span>)</span> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">ci</font> = <font color="Maroon" title="c4">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><span class="kw">assume </span><a NAME="E2:37_1_1_2"/>
<font color="Maroon" title="c5">i</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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</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="c4">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></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="c5">i</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"><font color="Maroon" title="c2">M</font></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="c5">i</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</font>, <font color="Maroon" title="c9">x3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:37_1_1_2"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:37_1_1_2"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E6:37_1_1_2"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c9">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E7:37_1_1_2"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T68" title="MCART_1:th.68">MCART_1:68</a></span>;<br/><span class="lab"><font color="Green" title="E32">A9</font></span>: <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c2">M</font></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="c5">i</font> = 
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c2">M</font></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="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_1_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c9">x3</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_1_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E5:37_1_1_2"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E6:37_1_1_2"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="altcat_1.html#D3" target="_self" title="ALTCAT_1:def.3">Def5</a></span>
;<br/><span class="lab"><font color="Green" title="E33">A10</font></span>: <font color="Maroon" title="c4">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<font color="Maroon" title="c4">c</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_1_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></span>
<br/>.= 
 <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E4:37_1_1_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E5:37_1_1_2"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E6:37_1_1_2"><span class="lab"><font color="Green" title="E30">A7</font></span></a></span>
;<br/><a NAME="E10:37_1_1_2"/>
( <font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>) &amp; <font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E4:37_1_1_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E5:37_1_1_2"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E6:37_1_1_2"><span class="lab"><font color="Green" title="E30">A7</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A11</font></span>: <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></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="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_1_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E5:37_1_1_2"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E6:37_1_1_2"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="altcat_1.html#D4" target="_self" title="ALTCAT_1:def.4">Def6</a></span>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></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="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_1_1_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></span>
;<br/><div><span class="lab"><font color="Green" title="E35">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:37_1_1_2_4"/>
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></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="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c2">M</font></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="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:37_1_1_2_4"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</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="#E11:37_1_1_2"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">j1</font>, <font color="Maroon" title="c12">j2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:37_1_1_2_4"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c11">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>)
 <span class="kw">and </span><br/><a NAME="E6:37_1_1_2_4"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>)
 <span class="kw">and </span><br/><a NAME="E7:37_1_1_2_4"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">j1</font>,<font color="Maroon" title="c12">j2</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="#E3:37_1_1_2_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T84" title="ZFMISC_1:th.84">ZFMISC_1:84</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">j2</font> = <font color="Maroon" title="c12">j2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1_2_4"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="funct_2.html#T66" title="FUNCT_2:th.66">FUNCT_2:66</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">j1</font> = <font color="Maroon" title="c11">j1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1_2_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="funct_2.html#T66" title="FUNCT_2:th.66">FUNCT_2:66</a></span>;<br/><a NAME="E10:37_1_1_2_4"/>
<font color="Maroon" title="c14">j1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c9">x3</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1_2_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E6:37_1_1_2_4"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="altcat_1.html#T3" target="_self" title="ALTCAT_1:th.3">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E11:37_1_1_2_4"/>
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c2">M</font></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="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E4:37_1_1_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E6:37_1_1_2"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E8:37_1_1_2"><span class="lab"><font color="Green" title="E32">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E13:37_1_1_2"/>
<font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c9">x3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c9">x3</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E4:37_1_1_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E6:37_1_1_2"><span class="lab"><font color="Green" title="E30">A7</font></span></a></span>;<br/><a NAME="E14:37_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A16</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">ci</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"><font color="Maroon" title="c2">M</font></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="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_1_1_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E8:37_1_1_2"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="altcat_1.html#T12" target="_self" title="ALTCAT_1:th.12">Th14</a></span>;<br/><a NAME="E15:37_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">ci</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</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="#E9:37_1_1_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E16:37_1_1_2"/>
<font color="Maroon" title="c4">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></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="c5">i</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"><font color="Maroon" title="c2">M</font></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="c5">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1_1_2"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E14:37_1_1_2"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="txt" href="#E12:37_1_1_2"><span class="lab"><font color="Green" title="E35">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="c5">c</font> = <font color="Maroon" title="c4">c</font> as   <a href="altcat_1.html#NM3" title="ALTCAT_1:NM.3">BinComp</a> of <font color="Maroon" title="c2">M</font> <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">set </span><font color="Maroon" title="c6">C</font> =  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #);<br/>
<a NAME="E9:37_1_1"/>
 <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #) is  <a href="altcat_1.html#V10" title="ALTCAT_1:attr.10">pseudo-functional</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">o1</font>, <font color="Maroon" title="c8">o2</font>, <font color="Maroon" title="c9">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="altcat_1.html#D13" target="_self" title="ALTCAT_1:def.13">ALTCAT_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #) <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span>)</span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<a NAME="E1:37_1_1_3"/>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>) &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>;<br/>
<a NAME="E2:37_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A17</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">thus </span> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #) <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>) = 
 <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>)</span>)</span>)</span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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="#E2:37_1_1_3"><span class="lab"><font color="Green" title="E28">A17</font></span></a>, <a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</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">C</font> =  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #) as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a>   <a href="altcat_1.html#V10" title="ALTCAT_1:attr.10">pseudo-functional</a>   <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; (  for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c7">C</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E11:37_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</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">a1</font>, <font color="Olive" title="b2">a2</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c7">C</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c9">a2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c7">C</font>; <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#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">a2</font>)</span><br/>

<span class="kw">thus </span><a NAME="E12:37_1_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">a2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E26">A1</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>
