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<span class="kw">let </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font> be    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Q</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">P</font> <a href="pcs_0.html#K2" title="PCS_0:func.2">\/</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">Q</font> &amp; 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#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <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 <font color="Maroon" title="c3">P</font> <a href="pcs_0.html#K2" title="PCS_0:func.2">\/</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c4">Q</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">S</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">f</font> =  <a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>;<br/>
<a NAME="E1:131"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E2:131"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E3:131"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E4:131"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
<span class="p1">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>;<br/>
<a NAME="E5:131"/><span class="lab"><font color="Green" title="E49">A5</font></span>: 
<span class="p1">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>;<br/>
<a NAME="E6:131"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:131_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>)</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>) or <font color="Maroon" title="c7">x</font> <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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:131_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:131_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A7</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:131_1_1"/><span class="lab"><font color="Green" title="E52">A8</font></span>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D1" target="_self" title="PCS_0:def.1">Def1</a></span>;<br/>
<a NAME="E4:131_1_1"/><span class="lab"><font color="Green" title="E53">A9</font></span>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c2">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D1" target="_self" title="PCS_0:def.1">Def1</a></span>;<br/>
<a NAME="E5:131_1_1"/><span class="lab"><font color="Green" title="E54">A10</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E4:131"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E3:131_1_1"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E6:131_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Q</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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E5:131"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E4:131_1_1"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:131_1_1"/>
<font color="Maroon" title="c7">x</font> <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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:131_1_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E5:131_1_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E2:131_1"/>
<font color="Maroon" title="c7">x</font> <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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>)</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="E4:131_1"/><span class="lab"><font color="Green" title="E51">A11</font></span>: 
<font color="Maroon" title="c7">x</font> <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="E5:131_1"/><span class="lab"><font color="Green" title="E52">A12</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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></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="E7:131_1"/><span class="lab"><font color="Green" title="E53">A13</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:131_1"/><span class="lab"><font color="Green" title="E54">A14</font></span>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></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="#E5:131_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E9:131_1"/>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E7:131_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E10:131_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c1">z</font></span>)</span> or <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c2">j</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:131_1"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="ref" href="pcs_0.html#D1" target="_self" title="PCS_0:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:131_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:131"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E5:131"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E4:131_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:131"/><span class="lab"><font color="Green" title="E51">A15</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:131_2"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>) <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>)</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>) or <font color="Maroon" title="c7">x</font> <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#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:131_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <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#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:131_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A16</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">P</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">Q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:131_2_1"/><span class="lab"><font color="Green" title="E53">A17</font></span>: 
<span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c1">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D6" target="_self" title="PCS_0:def.6">Def6</a></span>;<br/>
<a NAME="E4:131_2_1"/><span class="lab"><font color="Green" title="E54">A18</font></span>: 
<span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c2">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#D6" target="_self" title="PCS_0:def.6">Def6</a></span>;<br/>
<a NAME="E5:131_2_1"/><span class="lab"><font color="Green" title="E55">A19</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">P</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#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:131"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E4:131"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E3:131_2_1"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E6:131_2_1"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">Q</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#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:131"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E5:131"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E4:131_2_1"><span class="lab"><font color="Green" title="E54">A18</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:131_2_1"/>
<font color="Maroon" title="c7">x</font> <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#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:131_2_1"><span class="lab"><font color="Green" title="E52">A16</font></span></a>, <a class="txt" href="#E5:131_2_1"><span class="lab"><font color="Green" title="E55">A19</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <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#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E2:131_2"/>
<font color="Maroon" title="c7">x</font> <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#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>)</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="E4:131_2"/><span class="lab"><font color="Green" title="E52">A20</font></span>: 
<font color="Maroon" title="c7">x</font> <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="E5:131_2"/><span class="lab"><font color="Green" title="E53">A21</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#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></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="E7:131_2"/><span class="lab"><font color="Green" title="E54">A22</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:131_2"/><span class="lab"><font color="Green" title="E55">A23</font></span>: 
<span class="p1">(<span class="default"><a href="pcs_0.html#K4" title="PCS_0:func.4">pcs-InternalRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></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="#E5:131_2"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E9:131_2"/>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:131"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E7:131_2"><span class="lab"><font color="Green" title="E54">A22</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E10:131_2"/><span class="kw">then </span>
( <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c1">z</font></span>)</span> or <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c2">j</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:131_2"><span class="lab"><font color="Green" title="E55">A23</font></span></a>, <a class="ref" href="pcs_0.html#D6" target="_self" title="PCS_0:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:131_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:131"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E5:131"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E4:131_2"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:131"/>
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:131_3"/>
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>) <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</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> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>)</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <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">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>) or <font color="Maroon" title="c7">x</font> <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> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:131_3_1"/>
<font color="Maroon" title="c7">x</font> <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">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <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> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:131_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A24</font></span>: 
( <font color="Maroon" title="c7">x</font> <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">P</font> or <font color="Maroon" title="c7">x</font> <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">Q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:131_3_1"/><span class="lab"><font color="Green" title="E53">A25</font></span>: 
<span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">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"><span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c1">z</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="E4:131_3_1"/><span class="lab"><font color="Green" title="E54">A26</font></span>: 
<span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</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"><span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c2">j</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="E5:131_3_1"/><span class="lab"><font color="Green" title="E55">A27</font></span>: 
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c3">P</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> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:131"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E4:131"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E3:131_3_1"><span class="lab"><font color="Green" title="E53">A25</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E6:131_3_1"/>
the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c4">Q</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> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:131"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E5:131"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E4:131_3_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:131_3_1"/>
<font color="Maroon" title="c7">x</font> <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> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:131_3_1"><span class="lab"><font color="Green" title="E52">A24</font></span></a>, <a class="txt" href="#E5:131_3_1"><span class="lab"><font color="Green" title="E55">A27</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <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> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> or <font color="Maroon" title="c7">x</font> <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">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E2:131_3"/>
<font color="Maroon" title="c7">x</font> <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> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <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">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>)</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="E4:131_3"/><span class="lab"><font color="Green" title="E52">A28</font></span>: 
<font color="Maroon" title="c7">x</font> <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="E5:131_3"/><span class="lab"><font color="Green" title="E53">A29</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> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></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="E7:131_3"/><span class="lab"><font color="Green" title="E54">A30</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:131_3"/><span class="lab"><font color="Green" title="E55">A31</font></span>: 
<span class="p1">(<span class="default"><a href="pcs_0.html#K7" title="PCS_0:func.7">pcs-ToleranceRels</a> <span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span></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="#E5:131_3"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E9:131_3"/>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:131"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E7:131_3"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E10:131_3"/><span class="kw">then </span>
( <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"><span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c1">z</font></span>)</span> or <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"><span class="p2">(<span class="default"><a href="pcs_0.html#K18" title="PCS_0:func.18">MSSet</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Maroon" title="c2">j</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:131_3"><span class="lab"><font color="Green" title="E55">A31</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">hence </span><a NAME="E11:131_3"/>
<font color="Maroon" title="c7">x</font> <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">H<sub>1</sub></font>(<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:131"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E5:131"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E4:131_3"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:131"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Q</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">P</font> <a href="pcs_0.html#K2" title="PCS_0:func.2">\/</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">Q</font> &amp; 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#K19" title="PCS_0:func.19">pcs-sum</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <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 <font color="Maroon" title="c3">P</font> <a href="pcs_0.html#K2" title="PCS_0:func.2">\/</a> the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c4">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:131"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="#E7:131"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="ref" href="pcs_0.html#D36" target="_self" title="PCS_0:def.36">Def36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
