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<span class="kw">set </span><font color="Maroon" title="c2">L</font> =  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c4">A2</font> be   <a href="lattices.html#V3" title="LATTICES:attr.3">strict</a>   <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</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"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font> &amp; (  for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</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 <br/> (  the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c3">A1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Olive" title="b2">C</font></span>)</span> &amp;  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c3">A1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Olive" title="b2">C</font></span>)</span> ) ) &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font> &amp; (  for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</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 <br/> (  the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c4">A2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Olive" title="b2">C</font></span>)</span> &amp;  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c4">A2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Olive" title="b2">C</font></span>)</span> ) ) implies <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59_1_2"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E2:59_1_2"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
 for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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 <br/> (  the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c3">A1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Olive" title="b2">B</font></span>)</span> &amp;  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c3">A1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Olive" title="b2">B</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E3:59_1_2"/><span class="lab"><font color="Green" title="E49">A5</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E4:59_1_2"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
 for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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 <br/> (  the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c4">A2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Olive" title="b2">B</font></span>)</span> &amp;  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c4">A2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Olive" title="b2">B</font></span>)</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"> <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">a3</font> =  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c6">a4</font> =  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c4">A2</font>, <font color="Maroon" title="c7">a1</font> =  the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c8">a2</font> =  the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c4">A2</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="normform.html#K8" title="NORMFORM:func.8">Normal_forms_on</a> <font color="Maroon" title="c1">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_2"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E3:59_1_2"><span class="lab"><font color="Green" title="E49">A5</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="59_1_2_1"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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="Maroon" title="c7">a1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</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" 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="c7">a1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>)</span><br/><span class="kw">thus </span><font color="Maroon" title="c7">a1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>) = 
 <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c10">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1_2"><span class="lab"><font color="Green" title="E48">A4</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">a2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_1_2"><span class="lab"><font color="Green" title="E50">A6</font></span></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><span class="kw">end;</span></div>
<a NAME="E7:59_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A7</font></span>: 
<font color="Maroon" title="c7">a1</font> <a href="binop_1.html#R8" title="BINOP_1:pred.8">=</a> <font color="Maroon" title="c8">a2</font>
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="59_1_2_2"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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="Maroon" title="c5">a3</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a4</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</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" 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="c5">a3</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a4</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>)</span><br/><span class="kw">thus </span><font color="Maroon" title="c5">a3</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>) = 
 <a href="normform.html#K9" title="NORMFORM:func.9">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="normform.html#K10" title="NORMFORM:func.10">^</a> <font color="Maroon" title="c10">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1_2"><span class="lab"><font color="Green" title="E48">A4</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">a4</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_1_2"><span class="lab"><font color="Green" title="E50">A6</font></span></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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:59_1_2"/>
<font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_1_2"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E3:59_1_2"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E7:59_1_2"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="binop_1.html#T2" onmouseover="rs('binop_1/T2')" onmouseout="rh()">BINOP_1:2</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/>


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