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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <font color="Maroon" title="c2">U1</font>,   <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <font color="Maroon" title="c2">U1</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> $1;<br/>
<a NAME="E1:16_1_1"/><i><font color="Green" title="E12">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <font color="Maroon" title="c2">U1</font>  ex <font color="Olive" title="b3">m</font> being   <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <font color="Maroon" title="c2">U1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">m</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <font color="Maroon" title="c2">U1</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">xx</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">yy</font> = <font color="Maroon" title="c4">y</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c2">U1</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c6">yy</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">xx</font> as    <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <font color="Maroon" title="c2">U1</font> <b>by </b><i><a class="ref" href="endalg.html#T11" target="_self" title="ENDALG:th.11">Th11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">m</font>
;<br/>

<b>thus </b><a NAME="E3:16_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c7">m</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:16_1_1"/>
 ex <font color="Olive" title="b1">B</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <font color="Maroon" title="c2">U1</font> st <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <font color="Maroon" title="c2">U1</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>,<font color="Olive" title="b1">B</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="endalg.html#E1:16_1_1"><i><font color="Green" title="E12">A1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:16_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <font color="Maroon" title="c2">U1</font> st <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <font color="Maroon" title="c2">U1</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b2">x</font>
 ;<br/>


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