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<span class="kw">let </span><font color="Maroon" title="c1">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">d</font>, <font color="Olive" title="b2">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">G</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b5">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b3">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b3">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Olive" title="b3">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Olive" title="b2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Olive" title="b3">F</font> &amp; <font color="Olive" title="b4">G</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b3">F</font> &amp; <font color="Olive" title="b4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">e</font>,<font color="Olive" title="b1">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e</font> holds <br/><font color="Olive" title="b4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b5">p</font></span>)</span>,<font color="Olive" title="b1">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">G</font> <a href="finseqop.html#K2" title="FINSEQOP:func.2">[:]</a> (<font color="Olive" title="b5">p</font>,<font color="Olive" title="b1">d</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">d</font>, <font color="Maroon" title="c3">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">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>, <font color="Olive" title="b2">G</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b3">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">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="Olive" title="b1">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">G</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c2">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> holds <br/><font color="Olive" title="b2">G</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="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b3">p</font></span>)</span>,<font color="Maroon" title="c2">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="finseqop.html#K2" title="FINSEQOP:func.2">[:]</a> (<font color="Olive" title="b3">p</font>,<font color="Maroon" title="c2">d</font>)</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="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">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">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c5">G</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c2">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> holds <br/><font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b1">p</font></span>)</span>,<font color="Maroon" title="c2">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseqop.html#K2" title="FINSEQOP:func.2">[:]</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">d</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">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="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c5">G</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c2">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> implies <font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c6">p</font></span>)</span>,<font color="Maroon" title="c2">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseqop.html#K2" title="FINSEQOP:func.2">[:]</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">d</font>)</span>)</span> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
( <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c4">F</font> )
 <span class="kw">and </span><br/><a NAME="E2:46"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">and </span><br/><a NAME="E3:46"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c2">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</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">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c6">p</font></span>)</span>,<font color="Maroon" title="c2">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseqop.html#K2" title="FINSEQOP:func.2">[:]</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">d</font>)</span>)</span></span><br/>

<a NAME="E4:46"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseqop.html#K2" title="FINSEQOP:func.2">[:]</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">d</font>)</span>)</span> &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_2.html#T84" onmouseover="rs('finseq_2/T84')" onmouseout="rh()">FINSEQ_2:84</a></span>;<br/>
<a NAME="E5:46"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseqop.html#K2" title="FINSEQOP:func.2">[:]</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">d</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseqop.html#K2" title="FINSEQOP:func.2">[:]</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">d</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c6">p</font></span>)</span>,<font color="Maroon" title="c2">d</font>) = 
<font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p2">(<span class="default"><a href="finsop_1.html#K3" title="FINSOP_1:func.3">findom</a> <font color="Maroon" title="c6">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c3">e</font>)</span>)</span>)</span>)</span>,<font color="Maroon" title="c2">d</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="setwop_2.html#D2" target="_self" onmouseover="rs('setwop_2/D2')" onmouseout="rh()">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="finsop_1.html#K3" title="FINSOP_1:func.3">findom</a> <font color="Maroon" title="c6">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p2">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c3">e</font>)</span>)</span>,<font color="Maroon" title="c2">d</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="setwop_2.html#T13" target="_self" onmouseover="rs('setwop_2/T13')" onmouseout="rh()">Th13</a></span>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="finsop_1.html#K3" title="FINSOP_1:func.3">findom</a> <font color="Maroon" title="c6">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseqop.html#K2" title="FINSEQOP:func.2">[:]</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">d</font>)</span>)</span>,<font color="Maroon" title="c3">e</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="setwop_2.html#E30"><span class="lab"><font color="Green" title="E13">Lm5</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="finseqop.html#K2" title="FINSEQOP:func.2">[:]</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">d</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="setwop_2.html#D2" target="_self" onmouseover="rs('setwop_2/D2')" onmouseout="rh()">Def2</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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