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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">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b3">T</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Olive" title="b2">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b4">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="b5">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="b5">u</font> <a href="binop_1.html#R7" title="BINOP_1:pred.7">is_distributive_wrt</a> <font color="Olive" title="b4">F</font> holds <br/><font color="Olive" title="b5">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Olive" title="b1">d</font>,<font color="Olive" title="b3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b5">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b3">T</font></span>)</span></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">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">T</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Olive" title="b1">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font><br/> for <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="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> holds <br/><font color="Olive" title="b4">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Olive" title="b2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b4">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b2">T</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">T</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <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">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="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> holds <br/><font color="Olive" title="b3">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b1">T</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">T</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c3">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <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">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">u</font> <a href="binop_1.html#R7" title="BINOP_1:pred.7">is_distributive_wrt</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b2">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c4">T</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">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="Olive" title="b1">u</font> <a href="binop_1.html#R7" title="BINOP_1:pred.7">is_distributive_wrt</a> <font color="Maroon" title="c5">F</font> holds <br/><font color="Olive" title="b1">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c4">T</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">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="c6">u</font> <a href="binop_1.html#R7" title="BINOP_1:pred.7">is_distributive_wrt</a> <font color="Maroon" title="c5">F</font> implies <font color="Maroon" title="c6">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c4">T</font></span>)</span> )</span><br/>











<b>assume </b><a NAME="E1:62"/>
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="c1">D</font> holds  <font color="Maroon" title="c6">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">d2</font></span>)</span>
 ; <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"> <font color="Maroon" title="c6">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c4">T</font></span>)</span></span><br/>

<b>hence </b><a NAME="E2:62"/>
<font color="Maroon" title="c6">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <font color="Maroon" title="c2">d</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="finseqop.html#K3" title="FINSEQOP:func.3">[;]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c4">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseqop.html#T50" target="_self" title="FINSEQOP:th.50">Th50</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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