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<b>let </b><font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c4">o2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:51_1_2"/><i><font color="Green" title="E32">A2</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font> holds  <font color="Maroon" title="c3">o1</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K2" title="AFVECT0:func.2">Padd</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>
 <b>and </b><br/><a NAME="E2:51_1_2"/><i><font color="Green" title="E33">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font> holds  <font color="Maroon" title="c4">o2</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K2" title="AFVECT0:func.2">Padd</a> <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">X</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font>;<br/>
<a NAME="E3:51_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon" title="c3">o1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<b>consider </b><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font><b> such that </b><br/><a NAME="E2:51_1_2_1"/><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c6">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="c7">x1</font>,<font color="Maroon" title="c8">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<font color="Maroon" title="c3">o1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c3">o1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>
<b>by </b><i><a class="txt" href="afvect0.html#E2:51_1_2_1"><i><font color="Green" title="E34">A4</font></i></a></i>
<br/>.= 
 <a href="afvect0.html#K2" title="AFVECT0:func.2">Padd</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>
<b>by </b><i><a class="txt" href="afvect0.html#E1:51_1_2"><i><font color="Green" title="E32">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">o2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>
<b>by </b><i><a class="txt" href="afvect0.html#E2:51_1_2"><i><font color="Green" title="E33">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">o2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="afvect0.html#E2:51_1_2_1"><i><font color="Green" title="E34">A4</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:51_1_2_1"/>
<font color="Maroon" title="c3">o1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:51_1_2"/>
<font color="Maroon" title="c3">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


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