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<span class="kw">let </span><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c3">C3</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</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><br/> for <font color="Olive" title="b3">h</font> being  <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c3">C3</font><br/> for <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">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="b5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C3</font> holds <br/> <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span>)</span>,<font color="Olive" title="b3">h</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">h</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">z</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">h</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">z</font></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">h</font> being  <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c3">C3</font><br/> for <font color="Olive" title="b2">x</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="b2">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="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C3</font> holds <br/> <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">z</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">z</font></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">h</font> be   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c3">C3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C3</font> holds <br/> <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">z</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">z</font></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C3</font> implies  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c7">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="E2:24"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">F</font> =  <a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">G</font> =  <a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">H</font> =  <a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font>;<br/>
<a NAME="E3:24"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_4.html#T1" target="_self" title="FUZZY_4:th.1">Th1</a></span>;<br/>
<a NAME="E4:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A3</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</a></span>;<br/>
<a NAME="E5:24"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
for <font color="Olive" title="b1">s</font> being  <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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">s</font> be   <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">s</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E3:24_1"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<font color="Maroon" title="c13">r</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:24_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:24_1"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon" title="c14">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:24_1"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E8:24_1"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E6:24_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font> = 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E6:24_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self" title="FUZZY_4:def.2">Def2</a></span>
<br/>.= 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</a></span>
<br/>.= 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T33" title="XXREAL_0:th.33">XXREAL_0:33</a></span>
<br/>.= 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E6:24_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self" title="FUZZY_4:def.2">Def2</a></span>
;<br/>
<a NAME="E10:24_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:24_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E6:24_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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="E6:24"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
for <font color="Olive" title="b1">s</font> being  <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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">s</font> be   <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">s</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E3:24_2"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c13">r</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:24_2"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:24_2"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c14">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:24_2"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_2"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E8:24_2"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E6:24_2"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font> = 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E6:24_2"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self" title="FUZZY_4:def.2">Def2</a></span>
<br/>.= 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_2"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</a></span>
<br/>.= 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T33" title="XXREAL_0:th.33">XXREAL_0:33</a></span>
<br/>.= 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E6:24_2"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self" title="FUZZY_4:def.2">Def2</a></span>
;<br/>
<a NAME="E10:24_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_2"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:24_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_2"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E6:24_2"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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="E7:24"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_4.html#T1" target="_self" title="FUZZY_4:th.1">Th1</a></span>;<br/>
<a NAME="E8:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A16</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</a></span>;<br/>
<a NAME="E9:24"/><span class="lab"><font color="Green" title="E20">A17</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c12">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> implies <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_3"/>
<font color="Maroon" title="c12">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</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">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:24_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:24_3"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</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="E10:24"/>
for <font color="Olive" title="b1">s</font> being  <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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">s</font> be   <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">s</font> implies <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_4"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:24_4"/><span class="lab"><font color="Green" title="E21">A18</font></span>: 
<font color="Maroon" title="c13">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:24_4"/><span class="lab"><font color="Green" title="E22">A19</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E18">A10</font></span></a></span>;<br/>
<a NAME="E5:24_4"/>
<font color="Maroon" title="c13">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="txt" href="#E3:24_4"><span class="lab"><font color="Green" title="E21">A18</font></span></a></span>;<br/>
<a NAME="E6:24_4"/><span class="kw">then </span>
<font color="Maroon" title="c13">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:24_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E20">A17</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:24_4"/>
<span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_4"><span class="lab"><font color="Green" title="E22">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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="E11:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A20</font></span>: 
 <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T59" title="XREAL_1:th.59">XREAL_1:59</a></span>;<br/>
<a NAME="E12:24"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_4.html#T1" target="_self" title="FUZZY_4:th.1">Th1</a></span>;<br/>
<a NAME="E13:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A21</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</a></span>;<br/>
<a NAME="E14:24"/><span class="lab"><font color="Green" title="E23">A22</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c12">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> implies <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_5"/>
<font color="Maroon" title="c12">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</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">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:24_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:24_5"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</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="E15:24"/>
for <font color="Olive" title="b1">s</font> being  <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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">s</font> be   <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">s</font> implies <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_6"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:24_6"/><span class="lab"><font color="Green" title="E24">A23</font></span>: 
<font color="Maroon" title="c13">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:24_6"/><span class="lab"><font color="Green" title="E25">A24</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E17">A4</font></span></a></span>;<br/>
<a NAME="E5:24_6"/>
<font color="Maroon" title="c13">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="txt" href="#E3:24_6"><span class="lab"><font color="Green" title="E24">A23</font></span></a></span>;<br/>
<a NAME="E6:24_6"/><span class="kw">then </span>
<font color="Maroon" title="c13">y</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> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:24_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:24"><span class="lab"><font color="Green" title="E23">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:24_6"/>
<span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_6"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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="E16:24"/><span class="kw">then </span>
 <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T59" title="XREAL_1:th.59">XREAL_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:24"/>
 <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T20" title="XXREAL_0:th.20">XXREAL_0:20</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>
