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<b>reconsider </b><font color="Maroon" title="c3">c</font> =  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>let </b><font color="Maroon" title="c4">A1</font>, <font color="Maroon" title="c5">A2</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#L2" title="ALTCAT_1:struct.2">AltCatStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> holds  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">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="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<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">A1</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b4">f</font> <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">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; <font color="Olive" title="b5">g</font> <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">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="c4">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> <font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b4">f</font> ) &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> holds  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c5">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="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<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="c5">A2</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b4">f</font> <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="c5">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; <font color="Olive" title="b5">g</font> <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="c5">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="c5">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> <font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b4">f</font> ) implies <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:12_1_2"/><i><font color="Green" title="E8">A19</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> holds  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">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="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<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">A1</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b4">f</font> <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">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; <font color="Olive" title="b5">g</font> <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">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="c4">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> <font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b4">f</font> ) )
 <b>and </b><br/><a NAME="E3:12_1_2"/><i><font color="Green" title="E9">A20</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> holds  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c5">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="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<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="c5">A2</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b4">f</font> <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="c5">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; <font color="Olive" title="b5">g</font> <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="c5">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="c5">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> <font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b4">f</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A2</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">AA1</font> = the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">A1</font>, <font color="Maroon" title="c7">AA2</font> = the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c5">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"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c3">c</font></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#E2:12_1_2"><i><font color="Green" title="E8">A19</font></i></a>, <a class="txt" href="msinst_1.html#E3:12_1_2"><i><font color="Green" title="E9">A20</font></i></a></i>;<br/>
<div><i><font color="Green" title="E10">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">AA1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">AA2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font></span><br/><b>thus </b><font color="Maroon" title="c6">AA1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font> = 
 <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>
<b>by </b><i><a class="txt" href="msinst_1.html#E2:12_1_2"><i><font color="Green" title="E8">A19</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">AA2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>
<b>by </b><i><a class="txt" href="msinst_1.html#E3:12_1_2"><i><font color="Green" title="E9">A20</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E6:12_1_2"/><b>then </b><i><font color="Green" title="E11">A22</font></i>: 
<font color="Maroon" title="c6">AA1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">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="c8">CC1</font> = the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c4">A1</font>, <font color="Maroon" title="c9">CC2</font> = the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c5">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"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c3">c</font></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#E2:12_1_2"><i><font color="Green" title="E8">A19</font></i></a>, <a class="txt" href="msinst_1.html#E3:12_1_2"><i><font color="Green" title="E9">A20</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">j</font>, <font color="Maroon" title="c12">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font> implies <font color="Maroon" title="c8">CC1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">CC2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> )</span><br/><b>assume </b><a NAME="E1:12_1_2_2"/><i><font color="Green" title="E12">A23</font></i>: 
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">c</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">CC1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">CC2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span><br/><b>then reconsider </b><font color="Maroon" title="c13">i'</font> = <font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c14">j'</font> = <font color="Maroon" title="c11">j</font>, <font color="Maroon" title="c15">k'</font> = <font color="Maroon" title="c12">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">c</font> ;<br/><b>reconsider </b><font color="Maroon" title="c16">I</font> = <font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c17">J</font> = <font color="Maroon" title="c11">j</font>, <font color="Maroon" title="c18">K</font> = <font color="Maroon" title="c12">k</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">A1</font> <b>by </b><i><a class="txt" href="msinst_1.html#E2:12_1_2"><i><font color="Green" title="E8">A19</font></i></a>, <a class="txt" href="msinst_1.html#E1:12_1_2_2"><i><font color="Green" title="E12">A23</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">I'</font> = <font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c20">J'</font> = <font color="Maroon" title="c11">j</font>, <font color="Maroon" title="c21">K'</font> = <font color="Maroon" title="c12">k</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c5">A2</font> <b>by </b><i><a class="txt" href="msinst_1.html#E3:12_1_2"><i><font color="Green" title="E9">A20</font></i></a>, <a class="txt" href="msinst_1.html#E1:12_1_2_2"><i><font color="Green" title="E12">A23</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c22">ijk</font> = <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>;<br/><a NAME="E5:12_1_2_2"/><i><font color="Green" title="E13">A24</font></i>: 
( <font color="Maroon" title="c8">CC1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; <font color="Maroon" title="c9">CC2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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="E6:12_1_2_2"/><i><font color="Green" title="E14">A25</font></i>: 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c3">c</font></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#E1:12_1_2_2"><i><font color="Green" title="E12">A23</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="E7:12_1_2_2"/>
<font color="Maroon" title="c8">CC1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">CC2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c23">Ci</font> = <font color="Maroon" title="c8">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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">AA1</font>,<font color="Maroon" title="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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#E2:12_1_2"><i><font color="Green" title="E8">A19</font></i></a>, <a class="txt" href="msinst_1.html#E6:12_1_2_2"><i><font color="Green" title="E14">A25</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="c24">Cj</font> = <font color="Maroon" title="c9">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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="c7">AA2</font>,<font color="Maroon" title="c7">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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="c7">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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#E3:12_1_2"><i><font color="Green" title="E9">A20</font></i></a>, <a class="txt" href="msinst_1.html#E6:12_1_2_2"><i><font color="Green" title="E14">A25</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="12_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="E1:12_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="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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="12_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_2_2_1_1_1"/><i><font color="Green" title="E15">A26</font></i>: 
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">CC1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">CC2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span><br/><div class="add"><a NAME="E2:12_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="c7">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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#E5:12_1_2"><i><font color="Green" title="E10">A21</font></i></a>, <a class="ref" href="altcat_1.html#T9" title="ALTCAT_1:th.9">ALTCAT_1:9</a></i>;<br/><a NAME="E3:12_1_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E16">A27</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">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="c6">AA1</font>,<font color="Maroon" title="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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="c24">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="c7">AA2</font>,<font color="Maroon" title="c7">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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#E1:12_1_2_2_1_1_1"><i><font color="Green" title="E15">A26</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="E4:12_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="c6">AA1</font>,<font color="Maroon" title="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> holds <br/><font color="Maroon" title="c23">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="c24">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="12_1_2_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">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">AA1</font>,<font color="Maroon" title="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> implies <font color="Maroon" title="c23">Ci</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">Cj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> )</span><br/>

<b>assume </b><a NAME="E1:12_1_2_2_1_1_1_1"/>
<font color="Maroon" title="c25">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">AA1</font>,<font color="Maroon" title="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c23">Ci</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">Cj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font></span><br/>

<a NAME="E2:12_1_2_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c25">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">AA1</font>,<font color="Maroon" title="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c25">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="c7">AA2</font>,<font color="Maroon" title="c7">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> )
 
<b>by </b><i><a class="txt" href="msinst_1.html#E6:12_1_2"><i><font color="Green" title="E11">A22</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="E3:12_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E17">A28</font></i>: 
( <font color="Maroon" title="c25">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">AA1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">AA1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c25">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="c7">AA2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">AA2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">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#E1:12_1_2_2"><i><font color="Green" title="E12">A23</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="E4:12_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E18">A29</font></i>: 
( <font color="Maroon" title="c25">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">AA1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c25">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">AA1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font> &amp; <font color="Maroon" title="c25">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="c7">AA2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c25">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="c7">AA2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">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="E5:12_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E19">A30</font></i>: 
( <font color="Maroon" title="c25">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="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c14">j'</font>,<font color="Maroon" title="c15">k'</font> &amp; <font color="Maroon" title="c25">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="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c13">i'</font>,<font color="Maroon" title="c14">j'</font> )
 
<b>by </b><i><a class="txt" href="msinst_1.html#E2:12_1_2"><i><font color="Green" title="E8">A19</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c26">M1</font>, <font color="Maroon" title="c27">N1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c28">f</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c26">M1</font>,<font color="Maroon" title="c27">N1</font><b> such that </b><br/><a NAME="E7:12_1_2_2_1_1_1_1"/><i><font color="Green" title="E20">A31</font></i>: 
( <font color="Maroon" title="c26">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i'</font> &amp; <font color="Maroon" title="c27">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j'</font> &amp; <font color="Maroon" title="c28">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c26">M1</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c27">N1</font> &amp; <font color="Maroon" title="c28">f</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c26">M1</font>,<font color="Maroon" title="c27">N1</font> )
 <b>by </b><i><a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">M2</font>, <font color="Maroon" title="c30">N2</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c31">g</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c29">M2</font>,<font color="Maroon" title="c30">N2</font><b> such that </b><br/><a NAME="E9:12_1_2_2_1_1_1_1"/><i><font color="Green" title="E21">A32</font></i>: 
( <font color="Maroon" title="c29">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j'</font> &amp; <font color="Maroon" title="c30">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k'</font> &amp; <font color="Maroon" title="c31">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c29">M2</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c30">N2</font> &amp; <font color="Maroon" title="c31">g</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c29">M2</font>,<font color="Maroon" title="c30">N2</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E5:12_1_2_2_1_1_1_1"><i><font color="Green" title="E19">A30</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<a NAME="E10:12_1_2_2_1_1_1_1"/><i><font color="Green" title="E22">A33</font></i>: 
<font color="Maroon" title="c25">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"><font color="Maroon" title="c31">g</font>,<font color="Maroon" title="c28">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E3:12_1_2_2_1_1_1_1"><i><font color="Green" title="E17">A28</font></i></a>, <a class="txt" href="msinst_1.html#E7:12_1_2_2_1_1_1_1"><i><font color="Green" title="E20">A31</font></i></a>, <a class="txt" href="msinst_1.html#E9:12_1_2_2_1_1_1_1"><i><font color="Green" title="E21">A32</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>then </b><font color="Maroon" title="c23">Ci</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">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="c4">A1</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font>,<font color="Maroon" title="c18">K</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c31">g</font>,<font color="Maroon" title="c28">f</font>
<b>by </b><i><a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c31">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c28">f</font>
<b>by </b><i><a class="txt" href="msinst_1.html#E2:12_1_2"><i><font color="Green" title="E8">A19</font></i></a>, <a class="txt" href="msinst_1.html#E4:12_1_2_2_1_1_1_1"><i><font color="Green" title="E18">A29</font></i></a>, <a class="txt" href="msinst_1.html#E7:12_1_2_2_1_1_1_1"><i><font color="Green" title="E20">A31</font></i></a>, <a class="txt" href="msinst_1.html#E9:12_1_2_2_1_1_1_1"><i><font color="Green" title="E21">A32</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="c5">A2</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c19">I'</font>,<font color="Maroon" title="c20">J'</font>,<font color="Maroon" title="c21">K'</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c31">g</font>,<font color="Maroon" title="c28">f</font>
<b>by </b><i><a class="txt" href="msinst_1.html#E3:12_1_2"><i><font color="Green" title="E9">A20</font></i></a>, <a class="txt" href="msinst_1.html#E4:12_1_2_2_1_1_1_1"><i><font color="Green" title="E18">A29</font></i></a>, <a class="txt" href="msinst_1.html#E7:12_1_2_2_1_1_1_1"><i><font color="Green" title="E20">A31</font></i></a>, <a class="txt" href="msinst_1.html#E9:12_1_2_2_1_1_1_1"><i><font color="Green" title="E21">A32</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c24">Cj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font>
<b>by </b><i><a class="txt" href="msinst_1.html#E10:12_1_2_2_1_1_1_1"><i><font color="Green" title="E22">A33</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="E12:12_1_2_2_1_1_1_1"/>
<font color="Maroon" title="c23">Ci</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">Cj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:12_1_2_2_1_1_1"/>
<font color="Maroon" title="c8">CC1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">CC2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E6:12_1_2"><i><font color="Green" title="E11">A22</font></i></a>, <a class="txt" href="msinst_1.html#E5:12_1_2_2"><i><font color="Green" title="E13">A24</font></i></a>, <a class="txt" href="msinst_1.html#E3:12_1_2_2_1_1_1"><i><font color="Green" title="E16">A27</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:12_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="c6">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="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">CC1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">CC2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span><br/><div class="add"><a NAME="E2:12_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="c23">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="c24">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#E6:12_1_2"><i><font color="Green" title="E11">A22</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><a NAME="E3:12_1_2_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c23">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="c24">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="E4:12_1_2_2_1_1_2"/>
<font color="Maroon" title="c8">CC1</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">CC2</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E5:12_1_2_2"><i><font color="Green" title="E13">A24</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:12_1_2"/>
<font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A2</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E2:12_1_2"><i><font color="Green" title="E8">A19</font></i></a>, <a class="txt" href="msinst_1.html#E3:12_1_2"><i><font color="Green" title="E9">A20</font></i></a>, <a class="txt" href="msinst_1.html#E6:12_1_2"><i><font color="Green" title="E11">A22</font></i></a>, <a class="ref" href="altcat_1.html#T10" title="ALTCAT_1:th.10">ALTCAT_1:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
