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<span class="kw">let </span><font color="Maroon" title="c3">P</font> be    <a href="pcs_0.html#L1" title="PCS_0:struct.1">TolStr</a> ; <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="c3">P</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  implies ( <font color="Maroon" title="c3">P</font> is  <a href="pcs_0.html#V4" title="PCS_0:attr.4">pcs-tol-reflexive</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="pcs_0.html#V5" title="PCS_0:attr.5">pcs-tol-irreflexive</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="pcs_0.html#V6" title="PCS_0:attr.6">pcs-tol-symmetric</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <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="c3">P</font> is  <a href="pcs_0.html#V4" title="PCS_0:attr.4">pcs-tol-reflexive</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="pcs_0.html#V5" title="PCS_0:attr.5">pcs-tol-irreflexive</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="pcs_0.html#V6" title="PCS_0:attr.6">pcs-tol-symmetric</a>  )</span><br/>

<a NAME="E2:32_1_1"/><span class="kw">then </span>
 <a href="pcs_0.html#G1" title="PCS_0:aggr.1">TolStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>, 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#R1" title="HIDDEN:pred.1">=</a>  <a href="pcs_0.html#K5" title="PCS_0:func.5">emptyTolStr</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcs_0.html#T2" target="_self" title="PCS_0:th.2">Th2</a></span>;<br/>
<a NAME="E3:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A2</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#R1" title="HIDDEN:pred.1">=</a>  <a href="pcs_0.html#K1" title="PCS_0:func.1">field</a>  the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c3">P</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:32_1_1"/>
 the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c3">P</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pcs_0.html#D9" target="_self" title="PCS_0:def.9">PCS_0:def 9</a> <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="c3">P</font> is  <a href="pcs_0.html#V5" title="PCS_0:attr.5">pcs-tol-irreflexive</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="pcs_0.html#V6" title="PCS_0:attr.6">pcs-tol-symmetric</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E5:32_1_1"/>
 the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c3">P</font> <a href="relat_2.html#R2" title="RELAT_2:pred.2">is_irreflexive_in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E3:32_1_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="relat_2.html#D10" title="RELAT_2:def.10">RELAT_2:def 10</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pcs_0.html#D10" target="_self" title="PCS_0:def.10">PCS_0:def 10</a> <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="c3">P</font> is  <a href="pcs_0.html#V6" title="PCS_0:attr.6">pcs-tol-symmetric</a> </span><br/>

<span class="kw">thus </span><a NAME="E6:32_1_1"/>
 the <a href="pcs_0.html#U1" title="PCS_0:sel.1">ToleranceRel</a> of <font color="Maroon" title="c3">P</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E3:32_1_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pcs_0.html#D11" target="_self" title="PCS_0:def.11">PCS_0:def 11</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/>


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