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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="prvect_1.html#K1" target="_self">.:</a> <font color="Olive">x'</font>,<font color="Olive">y'</font> );<br/>
<a NAME="E1:12_1_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> ;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">F</font> <a href="prvect_1.html#K1" target="_self">.:</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> ;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>take </b>
<font color="Maroon">x'</font>
;<br/>

<b>take </b>
<font color="Maroon">y'</font>
;<br/>

<b>thus </b><a NAME="E3:12_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="prvect_1.html#K1" target="_self">.:</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</font> being   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font><b> such that </b><br/><a NAME="E3:12_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>let </b><font color="Maroon">p1</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>, <font color="Maroon">p2</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">p1</font>, <font color="Maroon">y</font> = <font color="Maroon">p2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> ;<br/>
<a NAME="E5:12_1_1"/>
 ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp; <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="prvect_1.html#K1" target="_self">.:</a> <font color="Olive">x'</font>,<font color="Olive">y'</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:12_1_1"/>
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="prvect_1.html#K1" target="_self">.:</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/>


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