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<span class="kw">let </span><font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c3">C2</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a>   <a href="altcat_1.html#V13" title="ALTCAT_1:attr.13">quasi-discrete</a>   <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">C1</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b1">i</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C2</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b1">i</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) implies <font color="Maroon" title="c2">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:63_1_2"/><span class="lab"><font color="Green" title="E36">A19</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E2:63_1_2"/><span class="lab"><font color="Green" title="E37">A20</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">C1</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b1">i</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E3:63_1_2"/><span class="lab"><font color="Green" title="E38">A21</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E4:63_1_2"/><span class="lab"><font color="Green" title="E39">A22</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C2</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b1">i</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C2</font></span><br/>

<div><span class="lab"><font color="Green" title="E40">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c6">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">j</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">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:63_1_2_1"/><span class="lab"><font color="Green" title="E41">A24</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E2:63_1_2_1"/><span class="lab"><font color="Green" title="E42">A25</font></span>: 
( <font color="Maroon" title="c5">j</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">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">i2</font> = <font color="Maroon" title="c4">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E1:63_1_2_1"><span class="lab"><font color="Green" title="E41">A24</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">i1</font> = <font color="Maroon" title="c4">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E1:63_1_2_1"><span class="lab"><font color="Green" title="E41">A24</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_2_1_1"/>
( ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> ) or <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">j</font> or <font color="Maroon" title="c5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:63_1_2_1_1_1"/><span class="lab"><font color="Green" title="E43">A26</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><br/><div class="add"><a NAME="E2:63_1_2_1_1_1"/><span class="lab"><font color="Green" title="E44">A27</font></span>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c7">i2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">i2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C2</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c7">i2</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c7">i2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c7">i2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c7">i2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1_2"><span class="lab"><font color="Green" title="E39">A22</font></span></a></span>;<br/><a NAME="E3:63_1_2_1_1_1"/>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c8">i1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> &amp; the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C1</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c8">i1</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c8">i1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c8">i1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c8">i1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_2"><span class="lab"><font color="Green" title="E37">A20</font></span></a></span>;<br/><span class="kw">hence </span>the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">:-&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="c4">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_1_1_1"><span class="lab"><font color="Green" title="E43">A26</font></span></a>, <a class="ref" href="funcop_1.html#D10" title="FUNCOP_1:def.10">FUNCOP_1:def 10</a></span>
<br/>.= 
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_1_1_1"><span class="lab"><font color="Green" title="E43">A26</font></span></a>, <a class="txt" href="#E2:63_1_2_1_1_1"><span class="lab"><font color="Green" title="E44">A27</font></span></a>, <a class="ref" href="funcop_1.html#D10" title="FUNCOP_1:def.10">FUNCOP_1:def 10</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:63_1_2_1_1_2"/><span class="lab"><font color="Green" title="E43">A28</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">j</font> or <font color="Maroon" title="c5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c9">j1</font> = <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c10">k1</font> = <font color="Maroon" title="c6">k</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E2:63_1_2_1"><span class="lab"><font color="Green" title="E42">A25</font></span></a></span>;<br/><a NAME="E3:63_1_2_1_1_2"/><span class="lab"><font color="Green" title="E44">A29</font></span>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c10">k1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">j2</font> = <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c12">k2</font> = <font color="Maroon" title="c6">k</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E2:63_1_2_1"><span class="lab"><font color="Green" title="E42">A25</font></span></a></span>;<br/><a NAME="E5:63_1_2_1_1_2"/><span class="lab"><font color="Green" title="E45">A30</font></span>: 
( the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C2</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c11">j2</font>,<font color="Maroon" title="c12">k2</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">j2</font>,<font color="Maroon" title="c12">k2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c11">j2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c12">k2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C1</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c10">k1</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">j1</font>,<font color="Maroon" title="c10">k1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c10">k1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 ;<br/><a NAME="E6:63_1_2_1_1_2"/>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c11">j2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">j2</font>,<font color="Maroon" title="c12">k2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></span>;<br/><span class="kw">hence </span><a NAME="E7:63_1_2_1_1_2"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">C1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_2_1_1_2"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="txt" href="#E5:63_1_2_1_1_2"><span class="lab"><font color="Green" title="E45">A30</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">i2</font> = <font color="Maroon" title="c4">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">i1</font> = <font color="Maroon" title="c4">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_2_2_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j</font> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:63_1_2_2_1_1"/><span class="lab"><font color="Green" title="E41">A31</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">hence </span>the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> = 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c7">i1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>

<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="c4">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_2"><span class="lab"><font color="Green" title="E37">A20</font></span></a></span>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c6">i2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1_2"><span class="lab"><font color="Green" title="E39">A22</font></span></a></span>
<br/>.= 
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_2_1_1"><span class="lab"><font color="Green" title="E41">A31</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:63_1_2_2_1_2"/><span class="lab"><font color="Green" title="E41">A32</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c8">j2</font> = <font color="Maroon" title="c5">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">j1</font> = <font color="Maroon" title="c5">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">C1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a></span>;<br/><span class="kw">thus </span>the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> = 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c9">j1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>

<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_2_1_2"><span class="lab"><font color="Green" title="E41">A32</font></span></a>, <a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></span>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c8">j2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2_2_1_2"><span class="lab"><font color="Green" title="E41">A32</font></span></a>, <a class="ref" href="altcat_1.html#D20" target="_self" title="ALTCAT_1:def.20">Def20</a></span>
<br/>.= 
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E7:63_1_2"/><span class="kw">then </span>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E3:63_1_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="ref" href="altcat_1.html#T9" target="_self" title="ALTCAT_1:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:63_1_2"/>
<font color="Maroon" title="c2">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_2"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E3:63_1_2"><span class="lab"><font color="Green" title="E38">A21</font></span></a>, <a class="txt" href="#E5:63_1_2"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="ref" href="altcat_1.html#T10" target="_self" title="ALTCAT_1:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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