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<b>let </b><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#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">f</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">g</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">S</font> holds <br/>( ( <font color="Olive" title="b2">f</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> implies <font color="Olive" title="b3">g</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> ) &amp; ( <font color="Olive" title="b2">f</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> implies <font color="Olive" title="b3">g</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> ) &amp; ( <font color="Olive" title="b2">f</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> implies <font color="Olive" title="b3">g</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> ) )</span><br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c1">D</font><br/> for <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">S</font>  st <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> holds <br/>( ( <font color="Olive" title="b1">f</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> implies <font color="Olive" title="b2">g</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#V3" title="BINOP_1:attr.3">idempotent</a> implies <font color="Olive" title="b2">g</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> ) &amp; ( <font color="Olive" title="b1">f</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> implies <font color="Olive" title="b2">g</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> ) )</span><br/><b>let </b><font color="Maroon" title="c3">f</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> holds <br/>( ( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> implies <font color="Olive" title="b1">g</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> ) &amp; ( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> implies <font color="Olive" title="b1">g</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> ) &amp; ( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> implies <font color="Olive" title="b1">g</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> ) )</span><br/><b>let </b><font color="Maroon" title="c4">g</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> implies ( ( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> implies <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> ) &amp; ( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> implies <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> ) &amp; ( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> implies <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> ) ) )</span><br/>







<b>assume </b><a NAME="E1:1"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> implies <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> ) &amp; ( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> implies <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> ) &amp; ( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> implies <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> ) )</span><br/>

<a NAME="E2:1"/><i><font color="Green" title="E3">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c2">S</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E3:1"/>
( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> implies <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> implies <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> ) &amp; ( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> implies <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1"/><i><font color="Green" title="E4">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <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">D</font> holds  <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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> <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a></span><br/>

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">a'</font> = <font color="Maroon" title="c7">a'</font>, <font color="Maroon" title="c10">b'</font> = <font color="Maroon" title="c8">b'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> ;<br/>
<b>thus </b><font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> = 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="filter_2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="filter_2.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font>

<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c9">a'</font>
<b>by </b><i><a class="txt" href="filter_2.html#E1:1_1"><i><font color="Green" title="E4">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="filter_2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="filter_2.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:1"/>
( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> implies <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> implies <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_2"/><i><font color="Green" title="E4">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D15" title="BINOP_1:def.15">BINOP_1:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a></span><br/>

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D15" title="BINOP_1:def.15">BINOP_1:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font></span><br/>

<b>thus </b><font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> = 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="filter_2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="filter_2.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font>

<br/>.= 
<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="txt" href="filter_2.html#E1:1_2"><i><font color="Green" title="E4">A4</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E5:1"/><i><font color="Green" title="E4">A5</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span>,<font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D14" title="BINOP_1:def.14">BINOP_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">g</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a></span><br/>

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D14" title="BINOP_1:def.14">BINOP_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span>,<font color="Maroon" title="c7">c</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c8">a'</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c9">b'</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c10">c'</font> = <font color="Maroon" title="c7">c</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">a'</font> = <font color="Maroon" title="c8">a'</font>, <font color="Maroon" title="c12">b'</font> = <font color="Maroon" title="c9">b'</font>, <font color="Maroon" title="c13">c'</font> = <font color="Maroon" title="c10">c'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> ;<br/>
<b>thus </b><font color="Maroon" title="c4">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">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span>,<font color="Maroon" title="c7">c</font> = 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="filter_2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="filter_2.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<font color="Maroon" title="c7">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="filter_2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="filter_2.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span>,<font color="Maroon" title="c7">c</font>

<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c11">a'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c12">b'</font>,<font color="Maroon" title="c13">c'</font></span>)</span>
<b>by </b><i><a class="txt" href="filter_2.html#E5:1"><i><font color="Green" title="E4">A5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="filter_2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="filter_2.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>

<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font></span>)</span>
<b>by </b><i><a class="txt" href="filter_2.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="filter_2.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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