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<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being 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> <br/> for <font color="Olive" title="b2">a1</font>, <font color="Olive" title="b3">a2</font> being non <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">a2</font> holds <br/>for <font color="Olive" title="b4">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b5">R1</font>, <font color="Olive" title="b6">R2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Olive" title="b1">X</font>  st <font color="Olive" title="b5">R1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Olive" title="b4">f</font>,<font color="Olive" title="b2">a1</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  &amp; <font color="Olive" title="b6">R2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Olive" title="b4">f</font>,<font color="Olive" title="b3">a2</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  holds <br/> <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Olive" title="b5">R1</font>,<font color="Maroon" title="c1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Olive" title="b6">R2</font>,<font color="Maroon" title="c1">x</font></span><br/><b>let </b><font color="Maroon" title="c2">X</font> be  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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being non <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">a2</font> holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b4">R1</font>, <font color="Olive" title="b5">R2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b4">R1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">a1</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  &amp; <font color="Olive" title="b5">R2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">a2</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  holds <br/> <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Olive" title="b4">R1</font>,<font color="Maroon" title="c1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Olive" title="b5">R2</font>,<font color="Maroon" title="c1">x</font></span><br/><b>let </b><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">a2</font> be  non <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a2</font> implies for <font color="Olive" title="b1">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b2">R1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">a1</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  &amp; <font color="Olive" title="b3">R2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c4">a2</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  holds <br/> <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Olive" title="b2">R1</font>,<font color="Maroon" title="c1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Olive" title="b3">R2</font>,<font color="Maroon" title="c1">x</font> )</span><br/>





<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b2">R1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">a1</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  &amp; <font color="Olive" title="b3">R2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c4">a2</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  holds <br/> <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Olive" title="b2">R1</font>,<font color="Maroon" title="c1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Olive" title="b3">R2</font>,<font color="Maroon" title="c1">x</font></span><br/>

<b>let </b><font color="Maroon" title="c5">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">R1</font>, <font color="Olive" title="b2">R2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b1">R1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">a1</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  &amp; <font color="Olive" title="b2">R2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a2</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  holds <br/> <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Olive" title="b1">R1</font>,<font color="Maroon" title="c1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Olive" title="b2">R2</font>,<font color="Maroon" title="c1">x</font></span><br/><b>let </b><font color="Maroon" title="c6">R1</font>, <font color="Maroon" title="c7">R2</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">R1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">a1</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  &amp; <font color="Maroon" title="c7">R2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a2</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  implies  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">R1</font>,<font color="Maroon" title="c1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c7">R2</font>,<font color="Maroon" title="c1">x</font> )</span><br/>



<b>assume </b><a NAME="E2:29"/><i><font color="Green" title="E26">A2</font></i>: 
( <font color="Maroon" title="c6">R1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">a1</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  &amp; <font color="Maroon" title="c7">R2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">a2</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">R1</font>,<font color="Maroon" title="c1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c7">R2</font>,<font color="Maroon" title="c1">x</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">z1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">R1</font>,<font color="Maroon" title="c1">x</font> implies <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c7">R2</font>,<font color="Maroon" title="c1">x</font> )</span><br/><b>assume </b><a NAME="E1:29_1"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">R1</font>,<font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c7">R2</font>,<font color="Maroon" title="c1">x</font></span><br/><a NAME="E2:29_1"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c6">R1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">R2</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:29"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="taxonom1.html#E2:29"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="taxonom1.html#T10" target="_self" title="TAXONOM1:th.10">Th10</a>, <a class="ref" href="taxonom1.html#T19" target="_self" title="TAXONOM1:th.19">Th19</a></i>;<br/><a NAME="E3:29_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z1</font>,<font color="Maroon" title="c1">x</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="c6">R1</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:29_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><b>hence </b><a NAME="E4:29_1"/>
<font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c7">R2</font>,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E2:29_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</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>hence </b><a NAME="E4:29"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">R1</font>,<font color="Maroon" title="c1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c7">R2</font>,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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>
