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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> $1,$2,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> $1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<b>consider </b><font color="Maroon" title="c2">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" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E2:61_1_1"/><i><font color="Green" title="E37">A1</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" title="c1">A</font> holds  <font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</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" target="_self" title="ALTCAT_1:sch.2">ALTCAT_1:sch 2</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>,   <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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> $1,$2,<span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> $2,$3,<span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> $1</span>)</span>,<span class="p4">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> $1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> $1</span>)</span></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<b>consider </b><font color="Maroon" title="c3">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" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span><b> such that </b><br/><a NAME="E4:61_1_1"/><i><font color="Green" title="E38">A2</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" title="c1">A</font> holds  <font color="Maroon" title="c3">c</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</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" target="_self" title="ALTCAT_1:sch.4">ALTCAT_1:sch 4</a>();<br/></i>
<div><i><font color="Green" title="E39">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><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> ;<br/><b>assume </b><a NAME="E1:61_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">i1</font>, <font color="Maroon" title="c6">i2</font>, <font color="Maroon" title="c7">i3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:61_1_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c6">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c7">i3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <b>and </b><br/><a NAME="E4:61_1_1_1"/><i><font color="Green" title="E41">A5</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">i1</font>,<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">i3</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/><b>reconsider </b><font color="Maroon" title="c8">i1</font> = <font color="Maroon" title="c5">i1</font>, <font color="Maroon" title="c9">i2</font> = <font color="Maroon" title="c6">i2</font>, <font color="Maroon" title="c10">i3</font> = <font color="Maroon" title="c7">i3</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="altcat_1.html#E3:61_1_1_1"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_1_1_1"/>
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i3</font> ) or <font color="Maroon" title="c8">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i2</font> or <font color="Maroon" title="c9">i2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">i3</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:61_1_1_1_1_1"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font>
 <b>and </b><br/><a NAME="E2:61_1_1_1_1_1"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i3</font>
 ;<br/><div class="add"><i><font color="Green" title="E44">A8</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="c8">i1</font>,<font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">i3</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:61_1_1_1"><i><font color="Green" title="E41">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="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">i3</font>,<span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span>,<span class="p4">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_1.html#E4:61_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>
<br/>.= 
 <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">i3</font>,<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span>,<span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:61_1_1_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E2:61_1_1_1_1_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>
;<br/><i><font color="Green" title="E45">A9</font></i>: <font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c8">i1</font> = 
 <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c8">i1</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:61_1_1"><i><font color="Green" title="E37">A1</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>
;<br/><i><font color="Green" title="E46">A10</font></i>: <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> = 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c8">i1</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:61_1_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="altcat_1.html#E1:61_1_1_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="altcat_1.html#E2:61_1_1_1_1_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:61_1_1_1_1_1"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="altcat_1.html#D6" target="_self" title="ALTCAT_1:def.6">Def6</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span>,<span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T35" title="ZFMISC_1:th.35">ZFMISC_1:35</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span>,<span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>
;<br/><i><font color="Green" title="E47">A11</font></i>: <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="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="c2">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c8">i1</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:61_1_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="altcat_1.html#E1:61_1_1_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="altcat_1.html#E2:61_1_1_1_1_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:61_1_1_1_1_1"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="altcat_1.html#D5" target="_self" title="ALTCAT_1:def.5">Def5</a></i>
;<br/><a NAME="E7:61_1_1_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span>,<span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/><b>hence </b><a NAME="E8:61_1_1_1_1_1"/>
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is   <a href="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="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="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="c2">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="altcat_1.html#E3:61_1_1_1_1_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="altcat_1.html#E5:61_1_1_1_1_1"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="altcat_1.html#E6:61_1_1_1_1_1"><i><font color="Green" title="E47">A11</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>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_1_1_2"/><i><font color="Green" title="E42">A12</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i2</font> or <font color="Maroon" title="c9">i2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">i3</font> )
 ;<br/><div class="add"><div><i><font color="Green" title="E43">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i2</font> or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">i3</font> ) )
 </span><b>by </b><i><a class="txt" href="altcat_1.html#E1:61_1_1_1_1_2"><i><font color="Green" title="E42">A12</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E44">A14</font></i>: 
<font color="Maroon" title="c8">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i2</font>
 ;<br/><div class="add"><b>thus </b><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="c8">i1</font>,<font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">i3</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:61_1_1_1"><i><font color="Green" title="E41">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="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">i3</font>,<span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span>,<span class="p4">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_1.html#E4:61_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:61_1_1_1_1_2_1_1_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_2_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:61_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E44">A15</font></i>: 
<font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font>
 <b>and </b><br/><a NAME="E2:61_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E45">A16</font></i>: 
<font color="Maroon" title="c9">i2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">i3</font>
 ;<br/><div class="add"><b>thus </b><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="c8">i1</font>,<font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">i3</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:61_1_1_1"><i><font color="Green" title="E41">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="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">i3</font>,<span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span>,<span class="p4">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_1.html#E4:61_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>
<br/>.= 
 <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">i3</font>,<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span>,<span class="p3">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:61_1_1_1_1_2_1_1_2"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:61_1_1_1_1_2_1_1_2"><i><font color="Green" title="E45">A16</font></i></a>, <a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:61_1_1_1_1_2"/><i><font color="Green" title="E44">A17</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="altcat_1.html#E2:61_1_1"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/><a NAME="E4:61_1_1_1_1_2"/>
<font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">i3</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c9">i2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="altcat_1.html#E2:61_1_1"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/><a NAME="E5:61_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E45">A18</font></i>: 
( <font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="altcat_1.html#E1:61_1_1_1_1_2"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="altcat_1.html#E3:61_1_1_1_1_2"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>;<br/><i><font color="Green" title="E46">A19</font></i>: <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> = 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">i3</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:61_1_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
<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="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">i3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D6" target="_self" title="ALTCAT_1:def.6">Def6</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_1.html#E5:61_1_1_1_1_2"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>
;<br/><a NAME="E7:61_1_1_1_1_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><b>hence </b><a NAME="E8:61_1_1_1_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="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="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="c2">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="altcat_1.html#E2:61_1_1_1_1_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="altcat_1.html#E6:61_1_1_1_1_2"><i><font color="Green" title="E46">A19</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="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E6:61_1_1"/>
<font color="Maroon" title="c3">c</font> is <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2"><b>proof </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> ; <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><br/>

<b>assume </b><a NAME="E1:61_1_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">c</font>
 ;<br/>

<a NAME="E2:61_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:61_1_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="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 <b>by </b><i><a class="txt" href="altcat_1.html#E5:61_1_1"><i><font color="Green" title="E39">A3</font></i></a></i>;<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="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">c</font> = <font color="Maroon" title="c4">c</font> as   <a href="altcat_1.html#NM3" title="ALTCAT_1:NM.3">BinComp</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="txt" href="altcat_1.html#E5:61_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">C</font> =  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #);<br/>
<a NAME="E9:61_1_1"/>
 <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #) is <a href="altcat_1.html#V13" title="ALTCAT_1:attr.13">quasi-discrete</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">o1</font>, <font color="Maroon" title="c8">o2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">ALTCAT_1:def 20</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:61_1_1_3"/><i><font color="Green" title="E40">A20</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:61_1_1_3"/><i><font color="Green" title="E41">A21</font></i>: 
<font color="Maroon" title="c7">o1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">o2</font>
 ;<br/>

<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> = 
 <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c7">o1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:61_1_1"><i><font color="Green" title="E37">A1</font></i></a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:61_1_1_3"><i><font color="Green" title="E41">A21</font></i></a>, <a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:61_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="altcat_1.html#E1:61_1_1_3"><i><font color="Green" title="E40">A20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">C</font> =  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #) as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a> <a href="altcat_1.html#V13" title="ALTCAT_1:attr.13">quasi-discrete</a>  <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c7">C</font>
;<br/>

<b>thus </b><a NAME="E11:61_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">i</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c7">C</font>;<br/>

<b>thus </b><span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c8">i</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> = 
 <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c8">i</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:61_1_1"><i><font color="Green" title="E37">A1</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>
;<br/>


</div>
