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<span class="kw">let </span><font color="Maroon" title="c1">A</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"> for <font color="Olive" title="b1">K</font>, <font color="Olive" title="b2">L</font>, <font color="Olive" title="b3">M</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font> holds  <font color="Olive" title="b1">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Olive" title="b3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Olive" title="b2">L</font></span>)</span> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Olive" title="b3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c3">L</font>, <font color="Maroon" title="c4">M</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c4">M</font></span><br/>







<a NAME="E1:84"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
( <font color="Maroon" title="c3">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c2">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c2">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normform.html#T72" target="_self" title="NORMFORM:th.72">Th72</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E43">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">K</font>, <font color="Maroon" title="c6">L</font>, <font color="Maroon" title="c7">M</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c7">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c7">M</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c7">M</font></span>)</span> implies <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c7">M</font> )</span><br/><span class="kw">assume </span><a NAME="E1:84_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c7">M</font></span>)</span>
 ; <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="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c7">M</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c10">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:84_1_1"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font>
 <span class="kw">and </span><br/><a NAME="E4:84_1_1"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c7">M</font>
 <span class="kw">and </span><br/><a NAME="E5:84_1_1"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normform.html#T55" target="_self" title="NORMFORM:th.55">Th55</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">b1</font>, <font color="Maroon" title="c12">c1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E7:84_1_1"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<font color="Maroon" title="c11">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font>
 <span class="kw">and </span><br/><a NAME="E8:84_1_1"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
<font color="Maroon" title="c12">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 <span class="kw">and </span><br/><a NAME="E9:84_1_1"/><span class="lab"><font color="Green" title="E49">A8</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b1</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c12">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84_1_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="normform.html#T55" target="_self" title="NORMFORM:th.55">Th55</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">d</font> = <font color="Maroon" title="c9">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b1</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c12">c1</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84_1_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E9:84_1_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a></span>;<br/><a NAME="E11:84_1_1"/><span class="lab"><font color="Green" title="E50">A9</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b1</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c12">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c11">b1</font></span>)</span> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c12">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normform.html#T16" target="_self" title="NORMFORM:th.16">Th16</a></span>;<br/><a NAME="E12:84_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c11">b1</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normform.html#T26" target="_self" title="NORMFORM:th.26">Th26</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">c2</font> = <font color="Maroon" title="c9">b</font> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <font color="Maroon" title="c11">b1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="normform.html#K7" title="NORMFORM:func.7">DISJOINT_PAIRS</a> <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="normform.html#T45" target="_self" title="NORMFORM:th.45">Th45</a></span>;<br/><a NAME="E14:84_1_1"/>
<font color="Maroon" title="c14">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c6">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84_1_1"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="txt" href="#E7:84_1_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="normform.html#T56" target="_self" title="NORMFORM:th.56">Th56</a></span>;<br/><span class="kw">hence </span><a NAME="E15:84_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c7">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84_1_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E8:84_1_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="txt" href="#E9:84_1_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E11:84_1_1"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="normform.html#T56" target="_self" title="NORMFORM:th.56">Th56</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:84_1"/>
<font color="Maroon" title="c5">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c7">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c7">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="normform.html#E77"><span class="lab"><font color="Green" title="E28">Lm5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A10</font></span>: 
<font color="Maroon" title="c2">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c4">M</font>
 
;<br/>
<a NAME="E4:84"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c3">L</font></span>)</span> &amp; <font color="Maroon" title="c2">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c2">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normform.html#T72" target="_self" title="NORMFORM:th.72">Th72</a></span>;<br/>
<a NAME="E5:84"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c4">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c4">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E43">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:84"/>
<font color="Maroon" title="c2">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">K</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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>
