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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( not $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) );<br/>
<a NAME="E1:30_1_1"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z1</font>, <font color="Olive" title="b4">z2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b4">z2</font>] holds <br/><font color="Olive" title="b3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">z2</font>
 
;<br/>
<a NAME="E2:30_1_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">IT</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="c1">C1</font>,<font color="Maroon" title="c2">C2</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> <span class="kw"> such that </span><br/><a NAME="E4:30_1_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
( ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">IT</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font> &amp;  ex <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] ) ) ) &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">IT</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c3">IT</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S5" title="BINOP_1:sch.5">BINOP_1:sch 5</a>(<a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>);<br/></span>
<a NAME="E5:30_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">IT</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1: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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">IT</font> )</span><br/>

<a NAME="E1:30_1_1_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
( not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> implies  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> implies <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> implies <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )
 
;<br/>
<span class="kw">assume </span><a NAME="E2:30_1_1_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">IT</font></span><br/>

<a NAME="E3:30_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E4:30_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:30_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">IT</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:30_1_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E1:30_1_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></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="E6:30_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A7</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E7:30_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">c</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="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">IT</font> or <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1_1_2"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">IT</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="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><span class="kw"> such that </span><br/><a NAME="E3:30_1_1_2"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">IT</font>
 <span class="kw">and </span><br/><a NAME="E4:30_1_1_2"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:30_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font>
 <span class="kw">and </span><br/><a NAME="E7:30_1_1_2"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">and </span><br/><a NAME="E8:30_1_1_2"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E9:30_1_1_2"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c8">A</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D1" title="INTEGRA1:def.1">INTEGRA1:def 1</a></span>;<br/>
<a NAME="E2:30_1_1_2_1"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c8">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c8">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></span>;<br/>
<a NAME="E3:30_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A13</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T6" title="INTEGRA1:th.6">INTEGRA1:6</a></span>;<br/>
<a NAME="E4:30_1_1_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1_2_1"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="integra1.html#T6" title="INTEGRA1:th.6">INTEGRA1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:30_1_1_2_1"/>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1_2_1"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</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="E10:30_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A14</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y</font> implies <font color="Maroon" title="c3">IT</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:30_1_1_2"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E8:30_1_1_2"><span class="lab"><font color="Green" title="E22">A10</font></span></a></span>;<br/>
<a NAME="E11:30_1_1_2"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> implies <font color="Maroon" title="c3">IT</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:30_1_1_2"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E8:30_1_1_2"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E9:30_1_1_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:30_1_1_2"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1_2"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E8:30_1_1_2"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E10:30_1_1_2"><span class="lab"><font color="Green" title="E24">A14</font></span></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="E8:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">IT</font> is   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:30_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> st <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/>( ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> implies <font color="Olive">b<sub>1</sub></font> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">y</font> implies <font color="Olive">b<sub>1</sub></font> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E6:30_1_1"><span class="lab"><font color="Green" title="E19">A7</font></span></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>
