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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">I</font>, <font color="Olive" title="b2">J</font> being 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> <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="c1">D</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">I</font>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b6">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">J</font>,<font color="Maroon" title="c1">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> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  holds <br/>for <font color="Olive" title="b7">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font><br/> for <font color="Olive" title="b8">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">J</font> holds  <font color="Olive" title="b3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b7">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b8">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Olive" title="b4">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Olive" title="b5">f</font>,<font color="Olive" title="b6">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b7">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Olive" title="b4">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b5">f</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b8">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Olive" title="b6">g</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">I</font>, <font color="Maroon" title="c3">J</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">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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b4">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">J</font>,<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> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  holds <br/>for <font color="Olive" title="b5">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b6">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b5">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b6">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">g</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">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b5">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b3">f</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b6">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Olive" title="b4">g</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">F</font>, <font color="Maroon" title="c5">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" 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="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c1">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> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b4">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b4">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Olive" title="b1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b4">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Olive" title="b2">g</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c1">D</font>  st <font color="Maroon" title="c4">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b3">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">g</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">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b3">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Olive" title="b1">g</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">J</font>,<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="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> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  implies for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span> )</span><br/>









<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E15">A1</font></i>: 
( <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> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</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">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c8">G</font> = <font color="Maroon" title="c5">G</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c1">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">D</font> ;<br/>
<a NAME="E3:34"/><i><font color="Green" title="E16">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</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">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c10">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</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">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span></span><br/><a NAME="E1:34_1"/><i><font color="Green" title="E17">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">u</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><i><font color="Green" title="E18">A4</font></i>: <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> = 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T35" title="ZFMISC_1:th.35">ZFMISC_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font>
<b>by </b><i><a class="txt" href="matrix_3.html#E1:34"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="setwiseo.html#T26" title="SETWISEO:th.26">SETWISEO:26</a></i>
<br/>.= 
<font color="Maroon" title="c8">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E3:34"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="matrix_3.html#E1:34_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="finseqop.html#T82" title="FINSEQOP:th.82">FINSEQOP:82</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c12">z</font> = <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> ;<br/><a NAME="E5:34_1"/><i><font color="Green" title="E19">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">z</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_3.html#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font>

;<br/><a NAME="E7:34_1"/><b>then </b><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">z</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E5:34_1"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/><a NAME="E8:34_1"/><i><font color="Green" title="E21">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E9:34_1"/><i><font color="Green" title="E22">A8</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E10:34_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">z</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E11:34_1"/><b>then </b><i><font color="Green" title="E23">A9</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">z</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="matrix_3.html#E9:34_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:34_1"/><i><font color="Green" title="E24">A10</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c1">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E13:34_1"/><i><font color="Green" title="E25">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">z</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">z</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#T71" title="FUNCT_3:th.71">FUNCT_3:71</a></i>;<br/><a NAME="E14:34_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">z</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c1">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E8:34_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="matrix_3.html#E11:34_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E15:34_1"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">z</font></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E7:34_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="matrix_3.html#E12:34_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="matrix_3.html#E13:34_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E16:34_1"/><b>then </b><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c12">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#D4" title="FUNCOP_1:def.4">FUNCOP_1:def 4</a></i>;<br/><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
<b>by </b><i><a class="txt" href="matrix_3.html#E1:34"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="setwiseo.html#T26" title="SETWISEO:th.26">SETWISEO:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
<b>by </b><i><a class="txt" href="matrix_3.html#E1:34"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="setwiseo.html#T26" title="SETWISEO:th.26">SETWISEO:26</a></i>
<br/>.= 
<font color="Maroon" title="c8">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E16:34_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="funcop_1.html#T35" title="FUNCOP_1:th.35">FUNCOP_1:35</a></i>
;<br/><b>hence </b><a NAME="E18:34_1"/>
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</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">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E3:34_1"><i><font color="Green" title="E18">A4</font></i></a></i>; <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>
<b>hence </b><a NAME="E5:34"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> <font color="Maroon" title="c6">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font></span>)</span></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>
