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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="pcs_0.html#V17" title="PCS_0:attr.17">pcs-yielding</a>  implies ( <font color="Maroon" title="c3">f</font> is  <a href="waybel_3.html#V5" title="WAYBEL_3:attr.5">reflexive-yielding</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="pcs_0.html#V1" title="PCS_0:attr.1">transitive-yielding</a>  &amp; <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#V11" title="PCS_0:attr.11">pcs-tol-symmetric-yielding</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:97_3_1"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="pcs_0.html#V17" title="PCS_0:attr.17">pcs-yielding</a> 
 ; <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="c3">f</font> is  <a href="waybel_3.html#V5" title="WAYBEL_3:attr.5">reflexive-yielding</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="pcs_0.html#V1" title="PCS_0:attr.1">transitive-yielding</a>  &amp; <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#V11" title="PCS_0:attr.11">pcs-tol-symmetric-yielding</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:97_3_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="waybel_3.html#V5" title="WAYBEL_3:attr.5">reflexive-yielding</a> 
  <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="c3">f</font> is  <a href="pcs_0.html#V1" title="PCS_0:attr.1">transitive-yielding</a>  &amp; <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#V11" title="PCS_0:attr.11">pcs-tol-symmetric-yielding</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">y</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_3.html#D8" title="WAYBEL_3:def.8">WAYBEL_3:def 8</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="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">f</font> or <font color="Maroon" title="c4">y</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:97_3_1_1"/>
(  not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">f</font> or <font color="Maroon" title="c4">y</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_3_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="pcs_0.html#D29" target="_self" title="PCS_0:def.29">Def29</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">thus </span><a NAME="E3:97_3_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="pcs_0.html#V1" title="PCS_0:attr.1">transitive-yielding</a> 
  <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="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#V11" title="PCS_0:attr.11">pcs-tol-symmetric-yielding</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">y</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pcs_0.html#D4" target="_self" title="PCS_0:def.4">PCS_0:def 4</a> <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="c4">y</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">y</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:97_3_1_2"/>
( <font color="Maroon" title="c4">y</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">y</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_3_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="pcs_0.html#D29" target="_self" title="PCS_0:def.29">Def29</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">thus </span><a NAME="E4:97_3_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="97_3_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">y</font> be    <a href="pcs_0.html#L1" title="PCS_0:struct.1">TolStr</a> ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">y</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">y</font> is  <a href="pcs_0.html#V4" title="PCS_0:attr.4">pcs-tol-reflexive</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:97_3_1_3"/>
( <font color="Maroon" title="c4">y</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">y</font> is  <a href="pcs_0.html#V4" title="PCS_0:attr.4">pcs-tol-reflexive</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_3_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="pcs_0.html#D29" target="_self" title="PCS_0:def.29">Def29</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="c4">y</font> be    <a href="pcs_0.html#L1" title="PCS_0:struct.1">TolStr</a> ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">y</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">y</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="E5:97_3_1"/>
( <font color="Maroon" title="c4">y</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">y</font> is  <a href="pcs_0.html#V6" title="PCS_0:attr.6">pcs-tol-symmetric</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_3_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="pcs_0.html#D29" target="_self" title="PCS_0:def.29">Def29</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/>


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