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<span class="kw">let </span><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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e</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">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="c1">D</font><br/>  for <font color="Olive" title="b4">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b5">T1</font>, <font color="Olive" title="b6">T2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Olive" title="b4">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b2">F</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#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Olive" title="b2">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">e</font>,<font color="Olive" title="b1">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> &amp; (  for <font color="Olive" title="b7">d1</font>, <font color="Olive" title="b8">d2</font>, <font color="Olive" title="b9">d3</font>, <font color="Olive" title="b10">d4</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="Olive" title="b2">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<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="Olive" title="b7">d1</font>,<font color="Olive" title="b8">d2</font>)</span>)</span>,<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="Olive" title="b9">d3</font>,<font color="Olive" title="b10">d4</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b7">d1</font>,<font color="Olive" title="b9">d3</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b8">d2</font>,<font color="Olive" title="b10">d4</font>)</span>)</span>) ) holds <br/><font color="Olive" title="b3">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b5">T1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b6">T2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">G</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b5">T1</font>,<font color="Olive" title="b6">T2</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">e</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c1">D</font><br/>  for <font color="Olive" title="b3">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b4">T1</font>, <font color="Olive" title="b5">T2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Olive" title="b3">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">F</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#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c2">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> &amp; (  for <font color="Olive" title="b6">d1</font>, <font color="Olive" title="b7">d2</font>, <font color="Olive" title="b8">d3</font>, <font color="Olive" title="b9">d4</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="Olive" title="b1">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<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="Olive" title="b6">d1</font>,<font color="Olive" title="b7">d2</font>)</span>)</span>,<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="Olive" title="b8">d3</font>,<font color="Olive" title="b9">d4</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b6">d1</font>,<font color="Olive" title="b8">d3</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b7">d2</font>,<font color="Olive" title="b9">d4</font>)</span>)</span>) ) holds <br/><font color="Olive" title="b2">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b4">T1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b5">T2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b4">T1</font>,<font color="Olive" title="b5">T2</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font>, <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="c1">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">T1</font>, <font color="Olive" title="b3">T2</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>  st <font color="Maroon" title="c3">F</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#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c2">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> &amp; (  for <font color="Olive" title="b4">d1</font>, <font color="Olive" title="b5">d2</font>, <font color="Olive" title="b6">d3</font>, <font color="Olive" title="b7">d4</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="c4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b4">d1</font>,<font color="Olive" title="b5">d2</font>)</span>)</span>,<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="Olive" title="b6">d3</font>,<font color="Olive" title="b7">d4</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#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="b4">d1</font>,<font color="Olive" title="b6">d3</font>)</span>)</span>,<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="b5">d2</font>,<font color="Olive" title="b7">d4</font>)</span>)</span>) ) holds <br/><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="c3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b2">T1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b3">T2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b2">T1</font>,<font color="Olive" title="b3">T2</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T1</font>, <font color="Olive" title="b2">T2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c5">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>  st <font color="Maroon" title="c3">F</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#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c2">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> &amp; (  for <font color="Olive" title="b3">d1</font>, <font color="Olive" title="b4">d2</font>, <font color="Olive" title="b5">d3</font>, <font color="Olive" title="b6">d4</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="c4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b3">d1</font>,<font color="Olive" title="b4">d2</font>)</span>)</span>,<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="Olive" title="b5">d3</font>,<font color="Olive" title="b6">d4</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#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="b3">d1</font>,<font color="Olive" title="b5">d3</font>)</span>)</span>,<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="b4">d2</font>,<font color="Olive" title="b6">d4</font>)</span>)</span>) ) holds <br/><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="c3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b1">T1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Olive" title="b2">T2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Olive" title="b1">T1</font>,<font color="Olive" title="b2">T2</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">T1</font>, <font color="Maroon" title="c7">T2</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c5">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c2">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> &amp; (  for <font color="Olive" title="b1">d1</font>, <font color="Olive" title="b2">d2</font>, <font color="Olive" title="b3">d3</font>, <font color="Olive" title="b4">d4</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="c4">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>,<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="Olive" title="b3">d3</font>,<font color="Olive" title="b4">d4</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#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">d1</font>,<font color="Olive" title="b3">d3</font>)</span>)</span>,<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">d2</font>,<font color="Olive" title="b4">d4</font>)</span>)</span>) ) implies <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="c3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c6">T1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c7">T2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c6">T1</font>,<font color="Maroon" title="c7">T2</font>)</span>)</span> )</span><br/>













<a NAME="E1:40"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:40"/>
( <font color="Maroon" title="c3">F</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#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c2">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> &amp; (  for <font color="Olive" title="b1">d1</font>, <font color="Olive" title="b2">d2</font>, <font color="Olive" title="b3">d3</font>, <font color="Olive" title="b4">d4</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="c4">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>,<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="Olive" title="b3">d3</font>,<font color="Olive" title="b4">d4</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#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">d1</font>,<font color="Olive" title="b3">d3</font>)</span>)</span>,<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">d2</font>,<font color="Olive" title="b4">d4</font>)</span>)</span>) ) implies <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="c3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c6">T1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c7">T2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> (<font color="Maroon" title="c6">T1</font>,<font color="Maroon" title="c7">T2</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setwop_2.html#T32" target="_self" onmouseover="rs('setwop_2/T32')" onmouseout="rh()">Th32</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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