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<span class="kw">let </span><font color="Maroon" title="c1">D1</font>, <font color="Maroon" title="c2">D2</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">g1</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D1</font><br/> for <font color="Olive" title="b3">f2</font>, <font color="Olive" title="b4">g2</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font> holds <br/> ( ( <font color="Olive" title="b1">f1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b2">g1</font> &amp; <font color="Olive" title="b3">f2</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b4">g2</font> ) iff <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b3">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Olive" title="b2">g1</font>,<font color="Olive" title="b4">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">g1</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f2</font>, <font color="Olive" title="b2">g2</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font> holds <br/> ( ( <font color="Maroon" title="c3">f1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Olive" title="b1">f2</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b2">g2</font> ) iff <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Olive" title="b1">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Olive" title="b2">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f2</font>, <font color="Maroon" title="c6">g2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">f1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c6">g2</font> ) iff <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )</span><br/>











<span class="kw">thus </span><a NAME="E1:44"/>
( <font color="Maroon" title="c3">f1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c6">g2</font> implies <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> implies ( <font color="Maroon" title="c3">f1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c6">g2</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:44_1"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c3">f1</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c4">g1</font>
 <span class="kw">and </span><br/><a NAME="E2:44_1"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c3">f1</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Maroon" title="c4">g1</font>
 <span class="kw">and </span><br/><a NAME="E3:44_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c5">f2</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c6">g2</font>
 <span class="kw">and </span><br/><a NAME="E4:44_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c5">f2</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Maroon" title="c6">g2</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="binop_1.html#D11" title="BINOP_1:def.11">BINOP_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span></span><br/>

<span class="kw">thus </span><a NAME="E5:44_1"/>
( <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> &amp; <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:44_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E3:44_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E4:44_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="filter_1.html#T28" target="_self" title="FILTER_1:th.28">Th28</a>, <a class="ref" href="filter_1.html#T29" target="_self" title="FILTER_1:th.29">Th29</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="binop_1.html#D11" title="BINOP_1:def.11">BINOP_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:44"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span>
 <span class="kw">and </span><br/><a NAME="E3:44"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="binop_1.html#D11" title="BINOP_1:def.11">BINOP_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c6">g2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E4:44"/>
( <font color="Maroon" title="c3">f1</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c6">g2</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Maroon" title="c6">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="filter_1.html#T28" target="_self" title="FILTER_1:th.28">Th28</a>, <a class="ref" href="filter_1.html#T29" target="_self" title="FILTER_1:th.29">Th29</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="binop_1.html#D11" title="BINOP_1:def.11">BINOP_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
