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

<b>let </b><font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c4">S2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pcs_0.html#L1" title="PCS_0:struct.1">TolStr</a> ;<br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S1</font>;<br/><b>let </b><font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">S2</font>;<br/>





<b>set </b><font color="Maroon" title="c9">I1</font> = the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c3">S1</font>;<br/>
<b>set </b><font color="Maroon" title="c10">I2</font> = the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c4">S2</font>;<br/>
<b>set </b><font color="Maroon" title="c11">x</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<a NAME="E1:74"/><i><font color="Green" title="E27">A1</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E29"><i><font color="Green" title="E25">Lm1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E2:74"/>
(  not <span class="p1"><a href="pcs_0.html#K9" title="PCS_0:func.9">[^</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font></span><a href="pcs_0.html#K9" title="PCS_0:func.9">^]</a></span> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <span class="p1"><a href="pcs_0.html#K9" title="PCS_0:func.9">[^</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font></span><a href="pcs_0.html#K9" title="PCS_0:func.9">^]</a></span> or <font color="Maroon" title="c5">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">c</font> or <font color="Maroon" title="c7">b</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">d</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_1"/>
<span class="p1"><a href="pcs_0.html#K9" title="PCS_0:func.9">[^</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font></span><a href="pcs_0.html#K9" title="PCS_0:func.9">^]</a></span> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <span class="p1"><a href="pcs_0.html#K9" title="PCS_0:func.9">[^</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font></span><a href="pcs_0.html#K9" title="PCS_0:func.9">^]</a></span>
 ;<br/>

<a NAME="E2:74_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span>
 
<b>by </b><i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def3</a></i>;<br/>
<a NAME="E3:74_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c3">S1</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c4">S2</font> )
 
<b>by </b><i><a class="ref" href="pcs_0.html#D3" target="_self" title="PCS_0:def.3">Def1b</a></i>;<br/>
<b>hence </b><a NAME="E4:74_1"/>
( <font color="Maroon" title="c5">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">c</font> or <font color="Maroon" title="c7">b</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">d</font> )
 <b>by </b><i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def3</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:74"/>
( <font color="Maroon" title="c5">a</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">c</font> or <font color="Maroon" title="c7">b</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c8">d</font> )
 ;<br/>

<a NAME="E4:74"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p5"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p5"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p5"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p5"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c3">S1</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p5"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p5"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p5"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p5"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c4">S2</font> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E1:74"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def3</a></i>;<br/>
<b>hence </b><a NAME="E5:74"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:74"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="pcs_0.html#D3" target="_self" title="PCS_0:def.3">Def1b</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">PCS_0:def 7</a><br/>


</div>
