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<b>let </b><font color="Maroon" title="c3">f</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies ( <font color="Maroon" title="c3">f</font> is <a href="pcs_0.html#V9" title="PCS_0:attr.9">pcs-tol-reflexive-yielding</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="pcs_0.html#V10" title="PCS_0:attr.10">pcs-tol-irreflexive-yielding</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="pcs_0.html#V11" title="PCS_0:attr.11">pcs-tol-symmetric-yielding</a> ) )</span><br/>

<b>assume </b><a NAME="E1:49_1_1"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is <a href="pcs_0.html#V9" title="PCS_0:attr.9">pcs-tol-reflexive-yielding</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="pcs_0.html#V10" title="PCS_0:attr.10">pcs-tol-irreflexive-yielding</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="pcs_0.html#V11" title="PCS_0:attr.11">pcs-tol-symmetric-yielding</a> )</span><br/>

<b>thus </b><a NAME="E2:49_1_1"/>
<font color="Maroon" title="c3">f</font> is <a href="pcs_0.html#V9" title="PCS_0:attr.9">pcs-tol-reflexive-yielding</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is <a href="pcs_0.html#V10" title="PCS_0:attr.10">pcs-tol-irreflexive-yielding</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="pcs_0.html#V11" title="PCS_0:attr.11">pcs-tol-symmetric-yielding</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pcs_0.html#D16" target="_self" title="PCS_0:def.16">PCS_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> is    <a href="pcs_0.html#L1" title="PCS_0:struct.1">TolStr</a>  &amp; <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c4">i</font> is <a href="pcs_0.html#V4" title="PCS_0:attr.4">pcs-tol-reflexive</a> )</span><br/>

<b>thus </b><a NAME="E1:49_1_1_1"/>
( <font color="Maroon" title="c4">i</font> is    <a href="pcs_0.html#L1" title="PCS_0:struct.1">TolStr</a>  &amp; <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c4">i</font> is <a href="pcs_0.html#V4" title="PCS_0:attr.4">pcs-tol-reflexive</a> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:49_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:49_1_1"/>
<font color="Maroon" title="c3">f</font> is <a href="pcs_0.html#V10" title="PCS_0:attr.10">pcs-tol-irreflexive-yielding</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> is <a href="pcs_0.html#V11" title="PCS_0:attr.11">pcs-tol-symmetric-yielding</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pcs_0.html#D17" target="_self" title="PCS_0:def.17">PCS_0:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> is    <a href="pcs_0.html#L1" title="PCS_0:struct.1">TolStr</a>  &amp; <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c4">i</font> is <a href="pcs_0.html#V5" title="PCS_0:attr.5">pcs-tol-irreflexive</a> )</span><br/>

<b>thus </b><a NAME="E1:49_1_1_2"/>
( <font color="Maroon" title="c4">i</font> is    <a href="pcs_0.html#L1" title="PCS_0:struct.1">TolStr</a>  &amp; <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c4">i</font> is <a href="pcs_0.html#V5" title="PCS_0:attr.5">pcs-tol-irreflexive</a> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:49_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pcs_0.html#D18" target="_self" title="PCS_0:def.18">PCS_0:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> is    <a href="pcs_0.html#L1" title="PCS_0:struct.1">TolStr</a>  &amp; <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c4">i</font> is <a href="pcs_0.html#V6" title="PCS_0:attr.6">pcs-tol-symmetric</a> )</span><br/>

<b>thus </b><a NAME="E4:49_1_1"/>
( <font color="Maroon" title="c4">i</font> is    <a href="pcs_0.html#L1" title="PCS_0:struct.1">TolStr</a>  &amp; <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c4">i</font> is <a href="pcs_0.html#V6" title="PCS_0:attr.6">pcs-tol-symmetric</a> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:49_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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