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<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">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">d</font>, <font color="Olive" title="b3">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b4">F</font>, <font color="Olive" title="b5">G</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="b6">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="b4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Olive" title="b3">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="b4">F</font> &amp; <font color="Olive" title="b5">G</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b4">F</font> &amp; <font color="Olive" title="b5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">d</font>,<font color="Olive" title="b3">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e</font> holds <br/><font color="Olive" title="b5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">d</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Olive" title="b1">B</font>,<font color="Olive" title="b6">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Olive" title="b1">B</font>,<span class="p1">(<span class="default"><font color="Olive" title="b5">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Olive" title="b2">d</font>,<font color="Olive" title="b6">f</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <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">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="c2">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="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="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="b1">d</font>,<font color="Olive" title="b2">e</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> (<font color="Olive" title="b1">d</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b5">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Olive" title="b1">d</font>,<font color="Olive" title="b5">f</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">d</font>, <font color="Maroon" title="c5">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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>, <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="c2">D</font><br/>  for <font color="Olive" title="b3">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="Olive" title="b1">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c5">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="c4">d</font>,<font color="Maroon" title="c5">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> holds <br/><font color="Olive" title="b2">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b3">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Maroon" title="c4">d</font>,<font color="Olive" title="b3">f</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">F</font>, <font color="Maroon" title="c7">G</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="c6">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c5">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="c6">F</font> &amp; <font color="Maroon" title="c7">G</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> holds <br/><font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Maroon" title="c4">d</font>,<font color="Olive" title="b1">f</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c8">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="c6">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c5">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="c6">F</font> &amp; <font color="Maroon" title="c7">G</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> implies <font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>) )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:13"/><span class="lab"><font color="Green" title="E9">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>  &amp; <font color="Maroon" title="c6">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  )
 <span class="kw">and </span><br/><a NAME="E2:13"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c5">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="c6">F</font>
 <span class="kw">and </span><br/><a NAME="E3:13"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c6">F</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="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font>] <span class="kw">means </span><font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> ($1,<font color="Maroon" title="c8">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> ($1,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>);<br/>
<a NAME="E4:13"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 for <font color="Olive" title="b1">B9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B9</font>] &amp;  not <font color="Olive" title="b2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">B9</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">B9</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <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">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B9</font>] &amp;  not <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">B9</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/><span class="kw">let </span><font color="Maroon" title="c10">c</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B9</font>] &amp;  not <font color="Maroon" title="c10">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">B9</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:13_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c9">B9</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c9">B9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>)
 <span class="kw">and </span><br/><a NAME="E2:13_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
 not <font color="Maroon" title="c10">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">B9</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">S<sub>1</sub></font>[<font color="Maroon" title="c9">B9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c9">B9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c8">f</font>)</span>)</span>) = 
<font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c9">B9</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">c</font></span>)</span>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:13_1"><span class="lab"><font color="Green" title="E14">A6</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="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c9">B9</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">c</font></span>)</span>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="binop_1.html#T11" onmouseover="rs('binop_1/T11')" onmouseout="rh()">BINOP_1:11</a></span>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c9">B9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c8">f</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">c</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="funcop_1.html#T53" onmouseover="rs('funcop_1/T53')" onmouseout="rh()">FUNCOP_1:53</a></span>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">B9</font> <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="c10">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:13_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="setwop_2.html#T2" target="_self" onmouseover="rs('setwop_2/T2')" onmouseout="rh()">Th2</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">assume </span><a NAME="E5:13"/>
<font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>)</span><br/>

<span class="kw">then </span><font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c8">f</font>)</span>)</span>) = 
<font color="Maroon" title="c5">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="setwiseo.html#T31" onmouseover="rs('setwiseo/T31')" onmouseout="rh()">SETWISEO:31</a></span>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="setwiseo.html#T31" onmouseover="rs('setwiseo/T31')" onmouseout="rh()">SETWISEO:31</a></span>
;<br/>
<a NAME="E7:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">C</font>]
 
;<br/>
<a NAME="E8:13"/>
 for <font color="Olive" title="b1">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="setwiseo.html#S2" onmouseover="rs('setwiseo/S2')" onmouseout="rh()">SETWISEO:sch 2</a>(<a class="txt" href="#E7:13"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E12">A4</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:13"/>
<font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c8">f</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">[;]</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c8">f</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"> verum</span><br/>


</div>
