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<span class="kw">let </span><font color="Maroon" title="c3">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C</font> being  () <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c3">I</font><br/>  for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Olive" title="b1">C</font></span>)</span> holds <br/> ( <font color="Olive" title="b2">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b3">q</font> iff  ex <font color="Olive" title="b4">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b5">P</font> being    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b6">p9</font>, <font color="Olive" title="b7">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b5">P</font> st <br/>( <font color="Olive" title="b4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b6">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b7">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b6">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b7">q9</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">C</font> be  () <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c3">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">q</font> iff  ex <font color="Olive" title="b3">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b4">P</font> being    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b5">p9</font>, <font color="Olive" title="b6">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b4">P</font> st <br/>( <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b6">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b5">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b6">q9</font> ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c5">R</font> =  <a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c7">q</font> iff  ex <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">P</font> being    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b3">p9</font>, <font color="Olive" title="b4">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">P</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b4">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Olive" title="b3">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b4">q9</font> ) )</span><br/>

<a NAME="E1:114"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:114"/>
( <font color="Maroon" title="c6">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c7">q</font> implies  ex <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">P</font> being    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b3">p9</font>, <font color="Olive" title="b4">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">P</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b4">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Olive" title="b3">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b4">q9</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">P</font> being    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b3">p9</font>, <font color="Olive" title="b4">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">P</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b4">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Olive" title="b3">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b4">q9</font> ) implies <font color="Maroon" title="c6">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c7">q</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:114_1"/>
<font color="Maroon" title="c6">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c7">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">P</font> being    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b3">p9</font>, <font color="Olive" title="b4">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">P</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b4">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Olive" title="b3">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b4">q9</font> )</span><br/>

<a NAME="E2:114_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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">(<span class="default"><a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></span>;<br/>
<a NAME="E3:114_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D36" target="_self" title="PCS_0:def.36">Def36</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:114_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 <span class="kw">and </span><br/><a NAME="E6:114_1"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:114_1"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:114_1"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:114_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">I1</font> = <font color="Maroon" title="c3">I</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">A1</font> = <font color="Maroon" title="c4">C</font> as  () <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c10">I1</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">i1</font> = <font color="Maroon" title="c9">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">P</font> = <font color="Maroon" title="c11">A1</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c12">i1</font> as    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b2">p9</font>, <font color="Olive" title="b3">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">P</font> st <br/>( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Olive" title="b2">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b3">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Olive" title="b2">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b3">q9</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c13">P</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p9</font>, <font color="Olive" title="b2">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">P</font> st <br/>( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Olive" title="b1">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b2">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Olive" title="b1">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">q9</font> )</span><br/>

<a NAME="E14:114_1"/>
<font color="Maroon" title="c8">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A1</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c12">i1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:114_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="pcs_0.html#D20" target="_self" title="PCS_0:def.20">Def20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">p9</font> = <font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c15">q9</font> = <font color="Maroon" title="c7">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:114_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">p9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">P</font> st <br/>( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c14">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Olive" title="b1">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c14">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b1">q9</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c15">q9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c14">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c15">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c14">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c15">q9</font> )</span><br/>

<span class="kw">thus </span><a NAME="E16:114_1"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c14">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c15">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c14">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c15">q9</font> )</span><br/>

<span class="kw">thus </span><a NAME="E17:114_1"/>
( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c14">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c15">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c15">q9</font></span><br/>

<span class="kw">thus </span><a NAME="E18:114_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c15">q9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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="c13">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:114_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E9:114_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="pcs_0.html#D20" target="_self" title="PCS_0:def.20">Def20</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">PCS_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c8">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c9">P</font> being    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a> , <font color="Maroon" title="c10">p9</font>, <font color="Maroon" title="c11">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">P</font><span class="kw"> such that </span><a NAME="E4:114"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 <span class="kw">and </span><br/><a NAME="E5:114"/><span class="lab"><font color="Green" title="E46">A7</font></span>: 
<font color="Maroon" title="c9">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">and </span><br/><a NAME="E6:114"/><span class="lab"><font color="Green" title="E47">A8</font></span>: 
<font color="Maroon" title="c10">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font>
 <span class="kw">and </span><br/><a NAME="E7:114"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
<font color="Maroon" title="c11">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font>
 <span class="kw">and </span><br/><a NAME="E8:114"/><span class="lab"><font color="Green" title="E49">A10</font></span>: 
<font color="Maroon" title="c10">p9</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c11">q9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c7">q</font></span><br/>

<a NAME="E9:114"/><span class="lab"><font color="Green" title="E50">A11</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">p9</font>,<font color="Maroon" title="c11">q9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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="c9">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:114"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">Def7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">I1</font> = <font color="Maroon" title="c3">I</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114"><span class="lab"><font color="Green" title="E45">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">i1</font> = <font color="Maroon" title="c8">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">I1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114"><span class="lab"><font color="Green" title="E45">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">A1</font> = <font color="Maroon" title="c4">C</font> as  () <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c12">I1</font> ;<br/>
<a NAME="E13:114"/>
<span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c14">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c14">A1</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c13">i1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D20" target="_self" title="PCS_0:def.20">Def20</a></span>;<br/>
<a NAME="E14:114"/><span class="kw">then </span>
 the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c14">A1</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c13">i1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<a NAME="E15:114"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <font color="Maroon" title="c4">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:114"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E6:114"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E7:114"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E9:114"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:114"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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">(<span class="default"><a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D36" target="_self" title="PCS_0:def.36">Def36</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pcs_0.html#D7" target="_self" title="PCS_0:def.7">PCS_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
