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<b>set </b><font color="Maroon" title="c2">c</font> =  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font>;<br/>
<b>let </b><font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c4">A2</font> be  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#L2" title="ALTCAT_1:struct.2">AltCatStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1_1_2"/><i><font color="Green" title="E1">A17</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> holds  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">A1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">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="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">A1</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> holds <br/>for <font color="Olive" title="b4">f1</font>, <font color="Olive" title="b5">f2</font>, <font color="Olive" title="b6">g1</font>, <font color="Olive" title="b7">g2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">A1</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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">g1</font>,<font color="Olive" title="b7">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">A1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">A1</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">g1</font>,<font color="Olive" title="b7">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><span class="p2">(<span class="default"><font color="Olive" title="b6">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b4">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b5">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 <b>and </b><br/><a NAME="E2:1_1_2"/><i><font color="Green" title="E2">A18</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> holds  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">A2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">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="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">A2</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> holds <br/>for <font color="Olive" title="b4">f1</font>, <font color="Olive" title="b5">f2</font>, <font color="Olive" title="b6">g1</font>, <font color="Olive" title="b7">g2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">A2</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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">g1</font>,<font color="Olive" title="b7">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">A2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c4">A2</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">g1</font>,<font color="Olive" title="b7">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><span class="p2">(<span class="default"><font color="Olive" title="b6">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b4">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b5">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">AA1</font> = the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">A1</font> as    <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"><span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</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="msinst_1.html#E1:1_1_2"><i><font color="Green" title="E1">A17</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">AA2</font> = the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">A2</font> as    <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"><span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</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="msinst_1.html#E2:1_1_2"><i><font color="Green" title="E2">A18</font></i></a></i>;<br/>
<div><i><font color="Green" title="E3">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</font> be    <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font>;<br/><b>thus </b><font color="Maroon" title="c5">AA1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> = 
 <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>
<b>by </b><i><a class="txt" href="msinst_1.html#E1:1_1_2"><i><font color="Green" title="E1">A17</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">AA2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>
<b>by </b><i><a class="txt" href="msinst_1.html#E2:1_1_2"><i><font color="Green" title="E2">A18</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E6:1_1_2"/><b>then </b><i><font color="Green" title="E4">A20</font></i>: 
<font color="Maroon" title="c5">AA1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">AA2</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#T9" title="ALTCAT_1:th.9">ALTCAT_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">CC1</font> = the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c8">CC2</font> = the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c4">A2</font> as    <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"><span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <b>by </b><i><a class="txt" href="msinst_1.html#E1:1_1_2"><i><font color="Green" title="E1">A17</font></i></a>, <a class="txt" href="msinst_1.html#E2:1_1_2"><i><font color="Green" title="E2">A18</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font>, <font color="Maroon" title="c11">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:1_1_2_2"/><i><font color="Green" title="E5">A21</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c12">i'</font> = <font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c13">j'</font> = <font color="Maroon" title="c10">j</font>, <font color="Maroon" title="c14">k'</font> = <font color="Maroon" title="c11">k</font> as    <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> ;<br/><b>reconsider </b><font color="Maroon" title="c15">I</font> = <font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c16">J</font> = <font color="Maroon" title="c10">j</font>, <font color="Maroon" title="c17">K</font> = <font color="Maroon" title="c11">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">A1</font> <b>by </b><i><a class="txt" href="msinst_1.html#E1:1_1_2"><i><font color="Green" title="E1">A17</font></i></a>, <a class="txt" href="msinst_1.html#E1:1_1_2_2"><i><font color="Green" title="E5">A21</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">I'</font> = <font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c19">J'</font> = <font color="Maroon" title="c10">j</font>, <font color="Maroon" title="c20">K'</font> = <font color="Maroon" title="c11">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">A2</font> <b>by </b><i><a class="txt" href="msinst_1.html#E2:1_1_2"><i><font color="Green" title="E2">A18</font></i></a>, <a class="txt" href="msinst_1.html#E1:1_1_2_2"><i><font color="Green" title="E5">A21</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c21">ijk</font> = <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>;<br/><a NAME="E5:1_1_2_2"/><i><font color="Green" title="E6">A22</font></i>: 
( <font color="Maroon" title="c7">CC1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">CC1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; <font color="Maroon" title="c8">CC2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">CC2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> )
 <b>by </b><i><a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>;<br/><a NAME="E6:1_1_2_2"/><i><font color="Green" title="E7">A23</font></i>: 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <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"><span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 <b>by </b><i><a class="txt" href="msinst_1.html#E1:1_1_2_2"><i><font color="Green" title="E5">A21</font></i></a>, <a class="ref" href="mcart_1.html#T73" title="MCART_1:th.73">MCART_1:73</a></i>;<br/><b>thus </b><a NAME="E7:1_1_2_2"/>
<font color="Maroon" title="c7">CC1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">CC2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c22">Ci</font> = <font color="Maroon" title="c7">CC1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> as   <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="c5">AA1</font>,<font color="Maroon" title="c5">AA1</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> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c5">AA1</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> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <b>by </b><i><a class="txt" href="msinst_1.html#E1:1_1_2"><i><font color="Green" title="E1">A17</font></i></a>, <a class="txt" href="msinst_1.html#E6:1_1_2_2"><i><font color="Green" title="E7">A23</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">Cj</font> = <font color="Maroon" title="c8">CC2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> as   <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="c6">AA2</font>,<font color="Maroon" title="c6">AA2</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> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c6">AA2</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> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <b>by </b><i><a class="txt" href="msinst_1.html#E2:1_1_2"><i><font color="Green" title="E2">A18</font></i></a>, <a class="txt" href="msinst_1.html#E6:1_1_2_2"><i><font color="Green" title="E7">A23</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_2_2_1_1"/>
( <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c5">AA1</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> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</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>  or <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c5">AA1</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> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_2_2_1_1_1"/><i><font color="Green" title="E8">A24</font></i>: 
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c5">AA1</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> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</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> 
 ;<br/><div class="add"><a NAME="E2:1_1_2_2_1_1_1"/><b>then </b>
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c6">AA2</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> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</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>by </b><i><a class="txt" href="msinst_1.html#E5:1_1_2"><i><font color="Green" title="E3">A19</font></i></a>, <a class="ref" href="altcat_1.html#T9" title="ALTCAT_1:th.9">ALTCAT_1:9</a></i>;<br/><a NAME="E3:1_1_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E9">A25</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">Ci</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">AA1</font>,<font color="Maroon" title="c5">AA1</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> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">Cj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c6">AA2</font>,<font color="Maroon" title="c6">AA2</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> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E1:1_1_2_2_1_1_1"><i><font color="Green" title="E8">A24</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:1_1_2_2_1_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">AA1</font>,<font color="Maroon" title="c5">AA1</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> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> holds <br/><font color="Maroon" title="c22">Ci</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Cj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_2_2_1_1_1_1"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">AA1</font>,<font color="Maroon" title="c5">AA1</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> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 ;<br/>

<a NAME="E2:1_1_2_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c5">AA1</font>,<font color="Maroon" title="c5">AA1</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="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c6">AA2</font>,<font color="Maroon" title="c6">AA2</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="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font> )
 
<b>by </b><i><a class="txt" href="msinst_1.html#E6:1_1_2"><i><font color="Green" title="E4">A20</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="E3:1_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E10">A26</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">AA1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">AA1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">AA2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">AA2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</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="msinst_1.html#E1:1_1_2_2"><i><font color="Green" title="E5">A21</font></i></a>, <a class="ref" href="altcat_1.html#D6" title="ALTCAT_1:def.6">ALTCAT_1:def 6</a></i>;<br/>
<a NAME="E4:1_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E11">A27</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AA1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c24">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AA1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c24">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">AA2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c24">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">AA2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<a NAME="E5:1_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E12">A28</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c13">j'</font>,<font color="Maroon" title="c14">k'</font> &amp; <font color="Maroon" title="c24">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c12">i'</font>,<font color="Maroon" title="c13">j'</font> )
 
<b>by </b><i><a class="txt" href="msinst_1.html#E1:1_1_2"><i><font color="Green" title="E1">A17</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c25">g1</font>, <font color="Maroon" title="c26">g2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:1_1_2_2_1_1_1_1"/><i><font color="Green" title="E13">A29</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <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="c25">g1</font>,<font color="Maroon" title="c26">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c25">g1</font>,<font color="Maroon" title="c26">g2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c13">j'</font>,<font color="Maroon" title="c14">k'</font> )
 <b>by </b><i><a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">f1</font>, <font color="Maroon" title="c28">f2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:1_1_2_2_1_1_1_1"/><i><font color="Green" title="E14">A30</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <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="c27">f1</font>,<font color="Maroon" title="c28">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c27">f1</font>,<font color="Maroon" title="c28">f2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c12">i'</font>,<font color="Maroon" title="c13">j'</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E5:1_1_2_2_1_1_1_1"><i><font color="Green" title="E12">A28</font></i></a>, <a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<a NAME="E10:1_1_2_2_1_1_1_1"/><i><font color="Green" title="E15">A31</font></i>: 
<font color="Maroon" title="c24">x</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">g1</font>,<font color="Maroon" title="c26">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">f1</font>,<font color="Maroon" title="c28">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E3:1_1_2_2_1_1_1_1"><i><font color="Green" title="E10">A26</font></i></a>, <a class="txt" href="msinst_1.html#E7:1_1_2_2_1_1_1_1"><i><font color="Green" title="E13">A29</font></i></a>, <a class="txt" href="msinst_1.html#E9:1_1_2_2_1_1_1_1"><i><font color="Green" title="E14">A30</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<font color="Maroon" title="c22">Ci</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">A1</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c15">I</font>,<font color="Maroon" title="c16">J</font>,<font color="Maroon" title="c17">K</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">g1</font>,<font color="Maroon" title="c26">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">f1</font>,<font color="Maroon" title="c28">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a>, <a class="txt" href="msinst_1.html#E10:1_1_2_2_1_1_1_1"><i><font color="Green" title="E15">A31</font></i></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"><font color="Maroon" title="c25">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c27">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c28">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="msinst_1.html#E1:1_1_2"><i><font color="Green" title="E1">A17</font></i></a>, <a class="txt" href="msinst_1.html#E4:1_1_2_2_1_1_1_1"><i><font color="Green" title="E11">A27</font></i></a>, <a class="txt" href="msinst_1.html#E7:1_1_2_2_1_1_1_1"><i><font color="Green" title="E13">A29</font></i></a>, <a class="txt" href="msinst_1.html#E9:1_1_2_2_1_1_1_1"><i><font color="Green" title="E14">A30</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c4">A2</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c18">I'</font>,<font color="Maroon" title="c19">J'</font>,<font color="Maroon" title="c20">K'</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">g1</font>,<font color="Maroon" title="c26">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">f1</font>,<font color="Maroon" title="c28">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="msinst_1.html#E2:1_1_2"><i><font color="Green" title="E2">A18</font></i></a>, <a class="txt" href="msinst_1.html#E4:1_1_2_2_1_1_1_1"><i><font color="Green" title="E11">A27</font></i></a>, <a class="txt" href="msinst_1.html#E7:1_1_2_2_1_1_1_1"><i><font color="Green" title="E13">A29</font></i></a>, <a class="txt" href="msinst_1.html#E9:1_1_2_2_1_1_1_1"><i><font color="Green" title="E14">A30</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c23">Cj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
<b>by </b><i><a class="txt" href="msinst_1.html#E10:1_1_2_2_1_1_1_1"><i><font color="Green" title="E15">A31</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E12:1_1_2_2_1_1_1_1"/>
<font color="Maroon" title="c22">Ci</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Cj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:1_1_2_2_1_1_1"/>
<font color="Maroon" title="c7">CC1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">CC2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E6:1_1_2"><i><font color="Green" title="E4">A20</font></i></a>, <a class="txt" href="msinst_1.html#E5:1_1_2_2"><i><font color="Green" title="E6">A22</font></i></a>, <a class="txt" href="msinst_1.html#E3:1_1_2_2_1_1_1"><i><font color="Green" title="E9">A25</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_2_2_1_1_2"/>
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c5">AA1</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> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:1_1_2_2_1_1_2"/><b>then </b>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c22">Ci</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c23">Cj</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="msinst_1.html#E6:1_1_2"><i><font color="Green" title="E4">A20</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><a NAME="E3:1_1_2_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c22">Ci</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c23">Cj</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="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/><b>hence </b><a NAME="E4:1_1_2_2_1_1_2"/>
<font color="Maroon" title="c7">CC1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">CC2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E5:1_1_2_2"><i><font color="Green" title="E6">A22</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:1_1_2"/>
<font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E1:1_1_2"><i><font color="Green" title="E1">A17</font></i></a>, <a class="txt" href="msinst_1.html#E2:1_1_2"><i><font color="Green" title="E2">A18</font></i></a>, <a class="txt" href="msinst_1.html#E6:1_1_2"><i><font color="Green" title="E4">A20</font></i></a>, <a class="ref" href="altcat_1.html#T10" title="ALTCAT_1:th.10">ALTCAT_1:10</a></i>;<br/>


</div>
