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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>] <b>means </b><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> $1,$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> $3,$2;<br/>
<a NAME="E1:37_1_1"/><i><font color="Green" title="E31">A2</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>



<b>set </b><font color="Maroon" title="c6">x</font> =  <a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c7">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E2:37_1_1_1"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="midsp_2.html#E1:37"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="midsp_2.html#D3" target="_self" title="MIDSP_2:def.3">Def3</a></i>;<br/>
<i><font color="Green" title="E33">A4</font></i>: <span class="p1">(<span class="default"><a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span> = 
 <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="ref" href="midsp_2.html#D6" target="_self" title="MIDSP_2:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="midsp_2.html#K3" title="MIDSP_2:func.3">Half</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">b</font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_2.html#E1:37"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="midsp_2.html#E2:37_1_1_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="midsp_2.html#D3" target="_self" title="MIDSP_2:def.3">Def3</a></i>
;<br/>
<b>take </b>
<font color="Maroon" title="c7">c</font>
;<br/>

<b>thus </b><a NAME="E4:37_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">c</font>]
 <b>by </b><i><a class="txt" href="midsp_2.html#E2:37_1_1_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="midsp_2.html#E3:37_1_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="rlvect_1.html#T21" title="RLVECT_1:th.21">RLVECT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">o</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E3:37_1_1"/><i><font color="Green" title="E32">A5</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c4">o</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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="midsp_2.html#E1:37_1_1"><i><font color="Green" title="E31">A2</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c4">o</font>
;<br/>

<b>thus </b><a NAME="E4:37_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span>,<font color="Olive" title="b2">b</font>
 <b>by </b><i><a class="txt" href="midsp_2.html#E3:37_1_1"><i><font color="Green" title="E32">A5</font></i></a></i>;<br/>


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