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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">d</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="b2">G</font>, <font color="Olive" title="b3">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="b4">u</font> being  <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <font color="Maroon" title="c1">D</font>  st <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="b3">F</font> &amp; <font color="Olive" title="b4">u</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Olive" title="b1">d</font> holds <br/><font color="Olive" title="b4">u</font> <a href="binop_1.html#R7" title="BINOP_1:pred.7">is_distributive_wrt</a> <font color="Olive" title="b3">F</font></span><br/><b>let </b><font color="Maroon" title="c2">d</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font>, <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">u</font> being  <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">G</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">u</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c2">d</font> holds <br/><font color="Olive" title="b3">u</font> <a href="binop_1.html#R7" title="BINOP_1:pred.7">is_distributive_wrt</a> <font color="Olive" title="b2">F</font></span><br/><b>let </b><font color="Maroon" title="c3">G</font>, <font color="Maroon" title="c4">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">u</font> being  <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <font color="Maroon" title="c1">D</font>  st <font color="Maroon" title="c3">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="Olive" title="b1">u</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c2">d</font> holds <br/><font color="Olive" title="b1">u</font> <a href="binop_1.html#R7" title="BINOP_1:pred.7">is_distributive_wrt</a> <font color="Maroon" title="c4">F</font></span><br/><b>let </b><font color="Maroon" title="c5">u</font> be   <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</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"> ( <font color="Maroon" title="c3">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">u</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">D</font></span>)</span>,<font color="Maroon" title="c2">d</font> implies <font color="Maroon" title="c5">u</font> <a href="binop_1.html#R7" title="BINOP_1:pred.7">is_distributive_wrt</a> <font color="Maroon" title="c4">F</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:65"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">F</font>
 <b>and </b><br/><a NAME="E2:65"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c5">u</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">D</font></span>)</span>,<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="c5">u</font> <a href="binop_1.html#R7" title="BINOP_1:pred.7">is_distributive_wrt</a> <font color="Maroon" title="c4">F</font></span><br/>

<b>let </b><font color="Maroon" title="c6">d1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D20" title="BINOP_1:def.20">BINOP_1:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></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="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">d1</font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">d2</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">d1</font>,<font color="Maroon" title="c7">d2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">d2</font></span>)</span></span><br/>



<b>thus </b><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">d1</font>,<font color="Maroon" title="c7">d2</font></span>)</span> = 
<font color="Maroon" title="c3">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"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">d1</font>,<font color="Maroon" title="c7">d2</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">d</font>
<b>by </b><i><a class="txt" href="finseqop.html#E2:65"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="funcop_1.html#T60" title="FUNCOP_1:th.60">FUNCOP_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c3">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="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">d1</font>,<font color="Maroon" title="c7">d2</font></span>)</span>,<font color="Maroon" title="c2">d</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c4">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">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">d1</font>,<font color="Maroon" title="c2">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">d2</font>,<font color="Maroon" title="c2">d</font></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E1:65"><i><font color="Green" title="E28">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="c4">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">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d1</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">d2</font>,<font color="Maroon" title="c2">d</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c4">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">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d1</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">d2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c4">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">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">d2</font></span>)</span>,<font color="Maroon" title="c2">d</font></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E2:65"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="funcop_1.html#T60" title="FUNCOP_1:th.60">FUNCOP_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c4">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">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">d2</font></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E2:65"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="funcop_1.html#T60" title="FUNCOP_1:th.60">FUNCOP_1:60</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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