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<b>let </b><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#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">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">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font><br/> for <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">G</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="b5">f</font>, <font color="Olive" title="b6">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="b3">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Olive" title="b3">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Olive" title="b3">F</font> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> &amp; <font color="Olive" title="b2">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="b3">F</font> &amp; <font color="Olive" title="b4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">e</font>,<font color="Olive" title="b2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">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="c2">D</font> holds  <font color="Olive" title="b3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">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="b4">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="b4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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="b3">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="b4">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b1">B</font>,<font color="Olive" title="b5">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b1">B</font>,<font color="Olive" title="b6">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">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"><font color="Olive" title="b4">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Olive" title="b5">f</font>,<font color="Olive" title="b6">f'</font></span>)</span></span><br/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c2">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="c2">D</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">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="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="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="c2">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="b6">d1</font>,<font color="Olive" title="b7">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="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="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="b6">d1</font>,<font color="Olive" title="b8">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="b7">d2</font>,<font color="Olive" title="b9">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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<font color="Olive" title="b4">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<font color="Olive" title="b5">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">f'</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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"> 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="c2">D</font><br/> for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">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="Maroon" title="c4">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="c4">e</font>,<font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e</font> &amp; ( for <font color="Olive" title="b5">d1</font>, <font color="Olive" title="b6">d2</font>, <font color="Olive" title="b7">d3</font>, <font color="Olive" title="b8">d4</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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="b5">d1</font>,<font color="Olive" title="b6">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="b7">d3</font>,<font color="Olive" title="b8">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="b5">d1</font>,<font color="Olive" title="b7">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="b6">d2</font>,<font color="Olive" title="b8">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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<font color="Olive" title="b3">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<font color="Olive" title="b4">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"><font color="Olive" title="b2">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">f'</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">F</font>, <font color="Maroon" title="c6">G</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <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="Maroon" title="c5">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> &amp; <font color="Maroon" title="c4">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="c5">F</font> &amp; <font color="Maroon" title="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c2">D</font> holds  <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">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="c6">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="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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="b3">d1</font>,<font color="Olive" title="b5">d3</font></span>)</span>,<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="b4">d2</font>,<font color="Olive" title="b6">d4</font></span>)</span> ) holds <br/><font color="Maroon" title="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<font color="Olive" title="b2">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">f'</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c8">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> &amp; <font color="Maroon" title="c4">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="c5">F</font> &amp; <font color="Maroon" title="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c2">D</font> holds  <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">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="c6">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="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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="b3">d3</font></span>)</span>,<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="b2">d2</font>,<font color="Olive" title="b4">d4</font></span>)</span> ) implies <font color="Maroon" title="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c8">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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"><font color="Maroon" title="c6">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">f'</font></span>)</span> )</span><br/>















<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c5">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c5">F</font> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c4">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="c5">F</font>
 <b>and </b><br/><a NAME="E4:10"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e</font>
 <b>and </b><br/><a NAME="E5:10"/><i><font color="Green" title="E11">A5</font></i>: 
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="c2">D</font> holds  <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">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="c6">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="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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="b3">d3</font></span>)</span>,<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="b2">d2</font>,<font color="Olive" title="b4">d4</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c8">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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"><font color="Maroon" title="c6">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">f'</font></span>)</span></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>] <b>means </b><font color="Maroon" title="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> $1,<font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> $1,<font color="Maroon" title="c8">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> $1,<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">f'</font></span>)</span>;<br/>
<a NAME="E6:10"/>
( <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c8">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e</font> )
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="#E2:10"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="#E3:10"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="setwiseo.html#T40" title="SETWISEO:th.40">SETWISEO:40</a></i>;<br/>
<a NAME="E7:10"/><b>then </b><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">C</font>]
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="#E2:10"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="#E3:10"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="#E4:10"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="setwiseo.html#T40" title="SETWISEO:th.40">SETWISEO:40</a></i>;<br/>
<a NAME="E8:10"/><i><font color="Green" title="E13">A7</font></i>: 
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">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B'</font>] &amp; not <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B'</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B'</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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 <font color="Maroon" title="c1">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B</font>] &amp; not <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/><b>let </b><font color="Maroon" title="c10">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B</font>] &amp; not <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>] )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:10_1"/><i><font color="Green" title="E14">A8</font></i>: 
<font color="Maroon" title="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c8">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">f'</font></span>)</span>
 <b>and </b><br/><a NAME="E2:10_1"/><i><font color="Green" title="E15">A9</font></i>: 
not <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/>

<b>set </b><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c7">f</font>;<br/>
<b>set </b><font color="Maroon" title="c12">s'</font> = <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c8">f'</font>;<br/>
<a NAME="E3:10_1"/>
( <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">c</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c8">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c8">f'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">c</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="#E2:10"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="#E2:10_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="setwop_2.html#T4" target="_self" title="SETWOP_2:th.4">Th4</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c8">f'</font></span>)</span> = 
<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">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c7">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c8">f'</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:10"><i><font color="Green" title="E11">A5</font></i></a></i>
<br/>.= 
<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">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c7">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c8">f'</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">f'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">c</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T48" title="FUNCOP_1:th.48">FUNCOP_1:48</a></i>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">f'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="#E2:10"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="#E1:10_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="#E2:10_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="setwop_2.html#T4" target="_self" title="SETWOP_2:th.4">Th4</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:10"/>
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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S2" title="SETWISEO:sch.2">SETWISEO:sch 2</a>(<a class="txt" href="#E7:10"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="#E8:10"><i><font color="Green" title="E13">A7</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:10"/>
<font color="Maroon" title="c6">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c8">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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"><font color="Maroon" title="c6">G</font> <a href="funcop_1.html#K6" title="FUNCOP_1:func.6">.:</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">f'</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
