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<b>let </b><font color="Maroon" title="c4">IT1</font>, <font color="Maroon" title="c5">IT2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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">S</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<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">S</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:16_1_2"/><i><font color="Green" title="E8">A15</font></i>: 
for <font color="Olive" title="b1">z1</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">S</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  ex <font color="Olive" title="b2">z2</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">S</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ex <font color="Olive" title="b3">s1</font>, <font color="Olive" title="b4">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b5">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> st <br/>( <font color="Olive" title="b2">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">IT1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z1</font> &amp; <font color="Olive" title="b4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">s1</font> &amp; <font color="Olive" title="b1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">s1</font>,<font color="Olive" title="b5">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b2">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b4">s2</font>,<font color="Olive" title="b5">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <b>and </b><br/><a NAME="E2:16_1_2"/><i><font color="Green" title="E9">A16</font></i>: 
for <font color="Olive" title="b1">z1</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">S</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  ex <font color="Olive" title="b2">z2</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">S</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ex <font color="Olive" title="b3">s1</font>, <font color="Olive" title="b4">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b5">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> st <br/>( <font color="Olive" title="b2">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z1</font> &amp; <font color="Olive" title="b4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">s1</font> &amp; <font color="Olive" title="b1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">s1</font>,<font color="Olive" title="b5">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b2">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b4">s2</font>,<font color="Olive" title="b5">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:16_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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">S</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">z1</font> = <font color="Maroon" title="c6">x</font> as    <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">S</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/><b>consider </b><font color="Maroon" title="c8">z12</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">S</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <font color="Maroon" title="c9">s11</font>, <font color="Maroon" title="c10">s12</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c11">z'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font><b> such that </b><br/><a NAME="E4:16_1_2_1"/><i><font color="Green" title="E10">A17</font></i>: 
( <font color="Maroon" title="c8">z12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">IT1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">z1</font> &amp; <font color="Maroon" title="c10">s12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">s11</font> &amp; <font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">s11</font>,<font color="Maroon" title="c11">z'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c8">z12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">s12</font>,<font color="Maroon" title="c11">z'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <b>by </b><i><a class="txt" href="group_10.html#E1:16_1_2"><i><font color="Green" title="E8">A15</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">z22</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">S</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <font color="Maroon" title="c13">s21</font>, <font color="Maroon" title="c14">s22</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c15">z''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font><b> such that </b><br/><a NAME="E6:16_1_2_1"/><i><font color="Green" title="E11">A18</font></i>: 
( <font color="Maroon" title="c12">z22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">z1</font> &amp; <font color="Maroon" title="c14">s22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">s21</font> &amp; <font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">s21</font>,<font color="Maroon" title="c15">z''</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c12">z22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">s22</font>,<font color="Maroon" title="c15">z''</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <b>by </b><i><a class="txt" href="group_10.html#E2:16_1_2"><i><font color="Green" title="E9">A16</font></i></a></i>;<br/><a NAME="E7:16_1_2_1"/><i><font color="Green" title="E12">A19</font></i>: 
( <font color="Maroon" title="c9">s11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s21</font> &amp; <font color="Maroon" title="c11">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">z''</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E4:16_1_2_1"><i><font color="Green" title="E10">A17</font></i></a>, <a class="txt" href="group_10.html#E6:16_1_2_1"><i><font color="Green" title="E11">A18</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>thus </b><a NAME="E8:16_1_2_1"/>
<font color="Maroon" title="c4">IT1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="group_10.html#E7:16_1_2_1"><i><font color="Green" title="E12">A19</font></i></a>, <a class="txt" href="group_10.html#E4:16_1_2_1"><i><font color="Green" title="E10">A17</font></i></a>, <a class="txt" href="group_10.html#E6:16_1_2_1"><i><font color="Green" title="E11">A18</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:16_1_2"/>
<font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>


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