<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">A1</font> be   <a href="binop_1.html#NM2">BinOp</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">A2</font> be   <a href="binop_1.html#NM2">BinOp</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>assume </b><a NAME="E1:1_1_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">z1</font>, <font color="Olive">z2</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span><br/> for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><br/> for <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font>  st <font color="Olive">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">y1</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x2</font>,<font color="Olive">y2</font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Maroon">A1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">z1</font>,<font color="Olive">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<b>assume </b><a NAME="E2:1_1_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">z1</font>, <font color="Olive">z2</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span><br/> for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><br/> for <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font>  st <font color="Olive">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">y1</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x2</font>,<font color="Olive">y2</font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Maroon">A2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">z1</font>,<font color="Olive">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y1</font>,<font color="Olive">y2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<a NAME="E3:1_1_2"/>
for <font color="Olive">z1</font>, <font color="Olive">z2</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds  <font color="Maroon">A1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">z1</font>,<font color="Olive">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">z1</font>,<font color="Olive">z2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z1</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">z2</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:1_1_2_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:1_1_2_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font>, <font color="Maroon">y2</font> = <font color="Maroon">y2</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">Y</font> <b>by </b><i>, </i>;<br/>
<a NAME="E7:1_1_2_1"/>
( <font color="Maroon">A1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">A2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E8:1_1_2_1"/>
<font color="Maroon">A1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_1_2"/>
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div>
