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<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  =  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>($1,$2),<font color="Maroon">F<sub>2</sub></font>($2,$3));<br/>
<span class="kw">consider </span><font color="Maroon" title="c1">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">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><span class="kw"> such that </span><br/><a NAME="E2:33_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c1">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> <font color="Maroon">F<sub>2</sub></font>(<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#S2" onmouseover="rs('altcat_1/S2')" onmouseout="rh()">ALTCAT_1:sch 2</a>();<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c2">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">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E4:33_1"/><span class="lab"><font color="Green" title="E15">A3</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c2">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#S4" onmouseover="rs('altcat_1/S4')" onmouseout="rh()">ALTCAT_1:sch 4</a>();<br/></span>
<a NAME="E5:33_1"/>
<font color="Maroon" title="c2">c</font> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funcop_1.html#D6" onmouseover="rs('funcop_1/D6')" onmouseout="rh()">FUNCOP_1:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c</font> or <font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">c</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> </span><br/>

<a NAME="E2:33_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font>, <font color="Maroon" title="c6">x3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:33_1_1"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c6">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 <span class="kw">and </span><br/><a NAME="E5:33_1_1"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T68" onmouseover="rs('mcart_1/T68')" onmouseout="rh()">MCART_1:68</a></span>;<br/>
<font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
<font color="Maroon" title="c2">c</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="multop_1.html#D1" onmouseover="rs('multop_1/D1')" onmouseout="rh()">MULTOP_1:def 1</a></span>
<br/>.= 
 <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font>))
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E4:33_1_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:33_1_1"/>
<font color="Maroon" title="c2">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">c</font> = <font color="Maroon" title="c2">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">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">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">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> holds <br/><font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="c1">M</font>,<font color="Maroon" title="c1">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="Olive" title="b1">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="c1">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="Olive" title="b1">i</font></span>)</span></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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> implies <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="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="c1">M</font>,<font color="Maroon" title="c1">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="c4">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="c1">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="c4">i</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:33_1_2"/>
<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">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="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="c1">M</font>,<font color="Maroon" title="c1">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="c4">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="c1">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="c4">i</font></span>)</span></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">object</a> <span class="kw"> such that </span><br/><a NAME="E3:33_1_2"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E4:33_1_2"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
<font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E5:33_1_2"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c7">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E6:33_1_2"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xtuple_0.html#K3" title="XTUPLE_0: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="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T68" onmouseover="rs('mcart_1/T68')" onmouseout="rh()">MCART_1:68</a></span>;<br/><a NAME="E7:33_1_2"/>
<font color="Maroon" title="c1">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:33_1_2"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E4:33_1_2"><span class="lab"><font color="Green" title="E17">A7</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A10</font></span>: <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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"><font color="Maroon" title="c1">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">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="#E2:33_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E4:33_1_2"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E18">A8</font></span></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="c1">M</font>,<font color="Maroon" title="c1">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="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="#E3:33_1_2"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E4:33_1_2"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="altcat_1.html#D4" onmouseover="rs('altcat_1/D4')" onmouseout="rh()">ALTCAT_1:def 4</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="c1">M</font>,<font color="Maroon" title="c1">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="c4">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1_2"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="multop_1.html#D1" onmouseover="rs('multop_1/D1')" onmouseout="rh()">MULTOP_1:def 1</a></span>
;<br/><span class="lab"><font color="Green" title="E21">A11</font></span>: <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c1">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="c4">i</font> = 
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c1">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="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="#E6:33_1_2"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="multop_1.html#D1" onmouseover="rs('multop_1/D1')" onmouseout="rh()">MULTOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x3</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_2"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E4:33_1_2"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="altcat_1.html#D3" onmouseover="rs('altcat_1/D3')" onmouseout="rh()">ALTCAT_1:def 3</a></span>
;<br/><div><span class="lab"><font color="Green" title="E22">A12</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_2_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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"><font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c1">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="c4">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>  implies <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c1">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="c4">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></a><div class="add"><span class="kw">assume </span><a NAME="E1:33_1_2_3"/>
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c1">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="c4">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c1">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="c4">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="c8">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:33_1_2_3"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
<font color="Maroon" title="c8">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"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_1_2"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">j1</font>, <font color="Maroon" title="c10">j2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:33_1_2_3"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>)
 <span class="kw">and </span><br/><a NAME="E6:33_1_2_3"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
<font color="Maroon" title="c10">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>)
 <span class="kw">and </span><br/><a NAME="E7:33_1_2_3"/>
<font color="Maroon" title="c8">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="c9">j1</font>,<font color="Maroon" title="c10">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="#E3:33_1_2_3"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T84" onmouseover="rs('zfmisc_1/T84')" onmouseout="rh()">ZFMISC_1:84</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">j2</font> = <font color="Maroon" title="c10">j2</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1_2_3"><span class="lab"><font color="Green" title="E25">A15</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">j1</font> = <font color="Maroon" title="c9">j1</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1_2_3"><span class="lab"><font color="Green" title="E24">A14</font></span></a></span>;<br/><a NAME="E10:33_1_2_3"/>
<font color="Maroon" title="c12">j1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">j2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x3</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:33_1_2"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E4:33_1_2"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E5:33_1_2_3"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="#E6:33_1_2_3"><span class="lab"><font color="Green" title="E25">A15</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:33_1_2_3"/>
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c1">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="c4">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="#E2:33_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:33_1_2"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E9:33_1_2"><span class="lab"><font color="Green" title="E21">A11</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="lab"><font color="Green" title="E23">A16</font></span>: <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="#E6:33_1_2"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="multop_1.html#D1" onmouseover="rs('multop_1/D1')" onmouseout="rh()">MULTOP_1:def 1</a></span>
<br/>.= 
 <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>),<font color="Maroon">F<sub>2</sub></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="#E4:33_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E3:33_1_2"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E4:33_1_2"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E18">A8</font></span></a></span>
;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">ci</font> = <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> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><a NAME="E13:33_1_2"/><span class="lab"><font color="Green" title="E24">A17</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">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"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:33_1_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/><a NAME="E14:33_1_2"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>))</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x3</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c9">F</font> =  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>));<br/>
<span class="kw">let </span><font color="Maroon" title="c10">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>))</span>)</span> or <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x3</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1_2_5"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>))</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x3</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:33_1_2_5"/><span class="lab"><font color="Green" title="E25">A18</font></span>: 
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>))</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:33_1_2_5"/><span class="lab"><font color="Green" title="E26">A19</font></span>: 
<font color="Maroon" title="c10">i</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> (<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>))</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">f</font>, <font color="Maroon" title="c13">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E6:33_1_2_5"/><span class="lab"><font color="Green" title="E27">A20</font></span>: 
<font color="Maroon" title="c11">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="c13">g</font>,<font color="Maroon" title="c12">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:33_1_2_5"/><span class="lab"><font color="Green" title="E28">A21</font></span>: 
<span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>))</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_2_5"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="ref" href="altcat_1.html#D9" onmouseover="rs('altcat_1/D9')" onmouseout="rh()">ALTCAT_1:def 9</a></span>;<br/>
<a NAME="E8:33_1_2_5"/>
( <font color="Maroon" title="c13">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>) &amp; <font color="Maroon" title="c12">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_2_5"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="txt" href="#E6:33_1_2_5"><span class="lab"><font color="Green" title="E27">A20</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:33_1_2_5"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x3</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:33_1_2"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E4:33_1_2"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E4:33_1_2_5"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E7:33_1_2_5"><span class="lab"><font color="Green" title="E28">A21</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E15:33_1_2"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">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="c1">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="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:33_1_2"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E5:33_1_2"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E11:33_1_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="txt" href="#E9:33_1_2"><span class="lab"><font color="Green" title="E21">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:33_1_2"/>
<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="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="c1">M</font>,<font color="Maroon" title="c1">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="c4">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="c1">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="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:33_1_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E8:33_1_2"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E10:33_1_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="altcat_1.html#NM3" title="ALTCAT_1:NM.3">BinComp</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D15" onmouseover="rs('pboole/D15')" onmouseout="rh()">PBOOLE:def 15</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">C</font> =  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">c</font> #);<br/>
<span class="kw">take </span>
 <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">c</font> #)
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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 <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">c</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; (  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> (  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">c</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> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) &amp; (  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font>, <font color="Olive" title="b5">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds   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">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">c</font> #) <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>,<font color="Olive" title="b5">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b4">j</font>,<font color="Olive" title="b5">k</font>)) ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E9:33_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">c</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> (  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">c</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> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) &amp; (  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font>, <font color="Olive" title="b5">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds   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">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">c</font> #) <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>,<font color="Olive" title="b5">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b4">j</font>,<font color="Olive" title="b5">k</font>)) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">c</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>) &amp; (  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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds   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">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">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>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font>)) ) )</span><br/>

<span class="kw">thus </span><a NAME="E10:33_1"/>
 the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">c</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds   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">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">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>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font>))</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">c</font> #) <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</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>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font>))</span><br/>

<span class="kw">thus </span><a NAME="E11:33_1"/>
 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">F<sub>1</sub></font>(),<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">c</font> #) <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</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>  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> (<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font>))
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
