<?xml version="1.0"?>
<div class="add">

<b>consider </b><font color="Maroon" title="c1">M</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.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><b> such that </b><br/><a NAME="E2:35_1"/><i><font color="Green" title="E19">A2</font></i>: 
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>)
 <b>from </b><i><a class="ref" href="altcat_1.html#S2" title="ALTCAT_1:sch.2">ALTCAT_1:sch 2</a>();<br/></i>
<b>deffunc </b><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> ) <b>-&gt; </b>   <a href="pboole.html#M1" title="PBOOLE:mode.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>2</sub></font>($2,$3),<font color="Maroon">F<sub>2</sub></font>($1,$2)</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> <font color="Maroon">F<sub>2</sub></font>($1,$2),<font color="Maroon">F<sub>2</sub></font>($2,$3);<br/>
<b>consider </b><font color="Maroon" title="c2">c</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.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><b> such that </b><br/><a NAME="E4:35_1"/><i><font color="Green" title="E20">A3</font></i>: 
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>)
 <b>from </b><i><a class="ref" href="altcat_1.html#S4" title="ALTCAT_1:sch.4">ALTCAT_1:sch 4</a>();<br/></i>
<a NAME="E5:35_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" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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#NK2" title="RELAT_1:NK.2">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#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<b>assume </b><a NAME="E1:35_1_1"/>
<font color="Maroon" title="c3">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="c2">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<a NAME="E2:35_1_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><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">set</a> <b> such that </b><br/><a NAME="E4:35_1_1"/><i><font color="Green" title="E21">A4</font></i>: 
( <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>() )
 <b>and </b><br/><a NAME="E5:35_1_1"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c3">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="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T72" title="MCART_1:th.72">MCART_1:72</a></i>;<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>
<b>by </b><i><a class="txt" href="#E5:35_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<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>)
<b>by </b><i><a class="txt" href="#E4:35_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="#E4:35_1_1"><i><font color="Green" title="E21">A4</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E7:35_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="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><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" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"><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="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><br/><b>assume </b><a NAME="E1:35_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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"><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="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><br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:35_1_2"/><i><font color="Green" title="E21">A6</font></i>: 
( <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>() &amp; <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>() &amp; <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>() )
 <b>and </b><br/><a NAME="E4:35_1_2"/><i><font color="Green" title="E22">A7</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="mcart_1.html#T72" title="MCART_1:th.72">MCART_1:72</a></i>;<br/><a NAME="E5:35_1_2"/><i><font color="Green" title="E23">A8</font></i>: 
<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>)
 <b>by </b><i><a class="txt" href="#E2:35_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="#E3:35_1_2"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/><i><font color="Green" title="E24">A9</font></i>: <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>
<b>by </b><i><a class="txt" href="#E4:35_1_2"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<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>)
<b>by </b><i><a class="txt" href="#E4:35_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="#E3:35_1_2"><i><font color="Green" title="E21">A6</font></i></a></i>
;<br/><b>then reconsider </b><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="E8:35_1_2"/><i><font color="Green" title="E25">A10</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 <b>by </b><i><a class="txt" href="#E6:35_1_2"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><i><font color="Green" title="E26">A11</font></i>: <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>
<b>by </b><i><a class="txt" href="#E2:35_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="#E3:35_1_2"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="#E5:35_1_2"><i><font color="Green" title="E23">A8</font></i></a></i>
<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>
<b>by </b><i><a class="txt" href="#E3:35_1_2"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="altcat_1.html#D6" title="ALTCAT_1:def.6">ALTCAT_1:def 6</a></i>
<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>
<b>by </b><i><a class="txt" href="#E4:35_1_2"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
;<br/><a NAME="E10:35_1_2"/><i><font color="Green" title="E27">A12</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">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" onclick="hs2(this)" href="javascript:()" title="35_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">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="c9">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/>

<b>set </b><font color="Maroon" title="c10">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/>
<b>assume </b><a NAME="E1:35_1_2_3"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">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/>

<b>then consider </b><font color="Maroon" title="c11">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_1_2_3"/><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><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>
 <b>and </b><br/><a NAME="E4:35_1_2_3"/><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon" title="c9">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>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><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><b> such that </b><br/><a NAME="E6:35_1_2_3"/><i><font color="Green" title="E30">A15</font></i>: 
<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>
 <b>and </b><br/><a NAME="E7:35_1_2_3"/><i><font color="Green" title="E31">A16</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E3:35_1_2_3"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="altcat_1.html#D11" title="ALTCAT_1:def.11">ALTCAT_1:def 11</a></i>;<br/>
<a NAME="E8:35_1_2_3"/>
 <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> <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="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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:35_1_2_3"/><b>then </b>
( <font color="Maroon" title="c13">g</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>) &amp; <font color="Maroon" title="c12">f</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>) )
 
<b>by </b><i><a class="txt" href="#E3:35_1_2_3"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="#E6:35_1_2_3"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E10:35_1_2_3"/>
<font color="Maroon" title="c9">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>)
 <b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E3:35_1_2"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="#E4:35_1_2_3"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="#E7:35_1_2_3"><i><font color="Green" title="E31">A16</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><i><font color="Green" title="E28">A17</font></i>: <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>
<b>by </b><i><a class="txt" href="#E4:35_1_2"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<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>
<b>by </b><i><a class="txt" href="#E3:35_1_2"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="altcat_1.html#D5" title="ALTCAT_1:def.5">ALTCAT_1:def 5</a></i>
;<br/><a NAME="E12:35_1_2"/><b>then </b><i><font color="Green" title="E29">A18</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">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>
 <b>by </b><i><a class="txt" href="#E2:35_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="#E3:35_1_2"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="#E6:35_1_2"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="#E10:35_1_2"><i><font color="Green" title="E27">A12</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_2_5"/>
<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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>then consider </b><font color="Maroon" title="c9">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_1_2_5"/><i><font color="Green" title="E30">A19</font></i>: 
<font color="Maroon" title="c9">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>
 <b>by </b><i><a class="txt" href="#E9:35_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">j1</font>, <font color="Maroon" title="c11">j2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:35_1_2_5"/><i><font color="Green" title="E31">A20</font></i>: 
<font color="Maroon" title="c10">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>)
 <b>and </b><br/><a NAME="E6:35_1_2_5"/><i><font color="Green" title="E32">A21</font></i>: 
<font color="Maroon" title="c11">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>)
 <b>and </b><br/><a NAME="E7:35_1_2_5"/>
<font color="Maroon" title="c9">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="c10">j1</font>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E3:35_1_2_5"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">j1</font> = <font color="Maroon" title="c10">j1</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E5:35_1_2_5"><i><font color="Green" title="E31">A20</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">j2</font> = <font color="Maroon" title="c11">j2</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E6:35_1_2_5"><i><font color="Green" title="E32">A21</font></i></a></i>;<br/><a NAME="E10:35_1_2_5"/>
<font color="Maroon" title="c12">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> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x3</font>)
 <b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E3:35_1_2"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="#E5:35_1_2_5"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="#E6:35_1_2_5"><i><font color="Green" title="E32">A21</font></i></a></i>;<br/><b>hence </b><a NAME="E11:35_1_2_5"/>
<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> 
 <b>by </b><i><a class="txt" href="#E2:35_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="#E3:35_1_2"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="#E11:35_1_2"><i><font color="Green" title="E28">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E14:35_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"><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="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>
 <b>by </b><i><a class="txt" href="#E8:35_1_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="#E9:35_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="#E12:35_1_2"><i><font color="Green" title="E29">A18</font></i></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#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><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> <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>set </b><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/>
<b>take </b>
 <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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E9:35_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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E10:35_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>)
 <b>by </b><i><a class="txt" href="#E2:35_1"><i><font color="Green" title="E19">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E11:35_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>)
 <b>by </b><i><a class="txt" href="#E4:35_1"><i><font color="Green" title="E20">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
