<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">D</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font>, <font color="Olive" title="b3">c3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b4">F</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Olive" title="b1">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c2</font> &amp; <font color="Olive" title="b1">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c3</font> &amp; <font color="Olive" title="b2">c2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c3</font> holds <br/><font color="Olive" title="b4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1"><a href="setwiseo.html#K4" title="SETWISEO:func.4">{.</a><span class="default"><font color="Olive" title="b1">c1</font>,<font color="Olive" title="b2">c2</font>,<font color="Olive" title="b3">c3</font></span><a href="setwiseo.html#K4" title="SETWISEO:func.4">.}</a></span>,<font color="Olive" title="b5">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p2">(<span class="default"><font color="Olive" title="b5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">c2</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">c3</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">c1</font>, <font color="Maroon" title="c4">c2</font>, <font color="Maroon" title="c5">c3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b1">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c3">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c2</font> &amp; <font color="Maroon" title="c3">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c3</font> &amp; <font color="Maroon" title="c4">c2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c3</font> holds <br/><font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1"><a href="setwiseo.html#K4" title="SETWISEO:func.4">{.</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c5">c3</font></span><a href="setwiseo.html#K4" title="SETWISEO:func.4">.}</a></span>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">c1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c2</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c3</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">F</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font>  st <font color="Maroon" title="c6">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c6">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c3">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c2</font> &amp; <font color="Maroon" title="c3">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c3</font> &amp; <font color="Maroon" title="c4">c2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c3</font> holds <br/><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1"><a href="setwiseo.html#K4" title="SETWISEO:func.4">{.</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c5">c3</font></span><a href="setwiseo.html#K4" title="SETWISEO:func.4">.}</a></span>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">c1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c2</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c3</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c7">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c6">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c3">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c2</font> &amp; <font color="Maroon" title="c3">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c3</font> &amp; <font color="Maroon" title="c4">c2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c3</font> implies <font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1"><a href="setwiseo.html#K4" title="SETWISEO:func.4">{.</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c5">c3</font></span><a href="setwiseo.html#K4" title="SETWISEO:func.4">.}</a></span>,<font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">c1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c2</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c3</font></span>)</span>) )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:3"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
( <font color="Maroon" title="c6">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c6">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  )
 <span class="kw">and </span><br/><a NAME="E2:3"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c3">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c3</font> or  not <font color="Maroon" title="c4">c2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c3</font> or <font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1"><a href="setwiseo.html#K4" title="SETWISEO:func.4">{.</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c5">c3</font></span><a href="setwiseo.html#K4" title="SETWISEO:func.4">.}</a></span>,<font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">c1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c2</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c3</font></span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E3:3"/>
( <font color="Maroon" title="c3">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c3</font> &amp; <font color="Maroon" title="c4">c2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c3</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1"><a href="setwiseo.html#K4" title="SETWISEO:func.4">{.</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c5">c3</font></span><a href="setwiseo.html#K4" title="SETWISEO:func.4">.}</a></span>,<font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">c1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c2</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c3</font></span>)</span>)</span><br/>

<a NAME="E4:3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A3</font></span>: 
 not <font color="Maroon" title="c5">c3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c4">c2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1"><a href="setwiseo.html#K4" title="SETWISEO:func.4">{.</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c5">c3</font></span><a href="setwiseo.html#K4" title="SETWISEO:func.4">.}</a></span>,<font color="Maroon" title="c7">f</font>) = 
<font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><span class="p2"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c4">c2</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c5">c3</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c7">f</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T3" onmouseover="rs('enumset1/T3')" onmouseout="rh()">ENUMSET1:3</a></span>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p2"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c4">c2</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<font color="Maroon" title="c7">f</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c3</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="setwop_2.html#T2" target="_self" onmouseover="rs('setwop_2/T2')" onmouseout="rh()">Th2</a></span>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">c1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c2</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c3</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="setwop_2.html#T1" target="_self" onmouseover="rs('setwop_2/T1')" onmouseout="rh()">Th1</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
