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<span class="kw">let </span><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#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">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</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="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>  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="b2">F</font> is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</a>  holds <br/><span class="p1">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Olive" title="b1">B</font>,<font color="Olive" title="b3">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Olive" title="b1">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">f</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</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> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b2">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>  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="Olive" title="b1">F</font> is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</a>  holds <br/><span class="p1">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b2">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">f</font></span>)</span>)</span><br/><span class="kw">let </span><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="c2">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> 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>  st <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</a>  holds <br/><span class="p1">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span> <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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f</font></span>)</span>)</span><br/><span class="kw">let </span><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" 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="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</a>  implies <span class="p1">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span> <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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font></span>)</span>) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
( <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  )
 <span class="kw">and </span><br/><a NAME="E2:19"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</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"> <span class="p1">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span> <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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font></span>)</span>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">e</font> =  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c4">F</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">u</font> =  <a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font>;<br/>
<a NAME="E3:19"/>
 <a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font> <a href="binop_1.html#R7" title="BINOP_1:pred.7">is_distributive_wrt</a> <font color="Maroon" title="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="finseqop.html#T63" onmouseover="rs('finseqop/T63')" onmouseout="rh()">FINSEQOP:63</a></span>;<br/>
<a NAME="E4:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 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> holds  <span class="p1">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span> <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="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="c4">F</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="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">d2</font></span>)</span>)
 
;<br/>
<a NAME="E5:19"/>
<span class="p1">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span> <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="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="finseqop.html#T61" onmouseover="rs('finseqop/T61')" onmouseout="rh()">FINSEQOP:61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:19"/>
<span class="p1">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span> <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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c5">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseqop.html#K5" title="FINSEQOP:func.5">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="setwop_2.html#T16" target="_self" onmouseover="rs('setwop_2/T16')" onmouseout="rh()">Th16</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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