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<b>let </b><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#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">d1</font>, <font color="Olive" title="b2">d2</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> 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><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>  st <font color="Olive" title="b4">F</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b5">G</font> holds <br/><font color="Olive" title="b4">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b3">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b5">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">d2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">d1</font>, <font color="Maroon" title="c4">d2</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" 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="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><br/> for <font color="Olive" title="b2">F</font>, <font color="Olive" title="b3">G</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b2">F</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b3">G</font> holds <br/><font color="Olive" title="b2">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">d2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c4">d2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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>  st <font color="Olive" title="b1">F</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">d1</font>,<font color="Maroon" title="c4">d2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d2</font></span>)</span></span><br/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">F</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c7">G</font> implies <font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<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="c3">d1</font>,<font color="Maroon" title="c4">d2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d2</font></span>)</span> )</span><br/>













<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c7">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<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="c3">d1</font>,<font color="Maroon" title="c4">d2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d2</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<span class="p2">(<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="c3">d1</font>,<font color="Maroon" title="c4">d2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">d1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<span class="p2">(<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="c3">d1</font>,<font color="Maroon" title="c4">d2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font> = 
<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="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></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="c3">d1</font>,<font color="Maroon" title="c4">d2</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T60" title="FUNCOP_1:th.60">FUNCOP_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c7">G</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="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c3">d1</font></span>)</span>,<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="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c4">d2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:46"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="binop_1.html#T23" title="BINOP_1:th.23">BINOP_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">d1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span>,<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="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c4">d2</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T60" title="FUNCOP_1:th.60">FUNCOP_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c7">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">d1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T60" title="FUNCOP_1:th.60">FUNCOP_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">d1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d2</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">c</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T48" title="FUNCOP_1:th.48">FUNCOP_1:48</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>hence </b><a NAME="E3:46"/>
<font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<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="c3">d1</font>,<font color="Maroon" title="c4">d2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">d2</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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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