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<b>let </b><font color="Maroon" title="c1">X</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">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <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">X</font>, <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <span class="p1"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">j</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be    <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> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <span class="p1"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <span class="p1"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <span class="p1"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j</font></span>)</span></span><br/>









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(  <a href="setwiseo.html#K9" title="SETWISEO:func.9">FinUnion</a> <font color="Maroon" title="c2">A</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> &amp;  <a href="setwiseo.html#K9" title="SETWISEO:func.9">FinUnion</a> <font color="Maroon" title="c2">A</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp;  <a href="setwiseo.html#K9" title="SETWISEO:func.9">FinUnion</a> <font color="Maroon" title="c2">A</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )
 
<b>by </b><i><a class="ref" href="setwiseo.html#T49" target="_self" title="SETWISEO:th.49">Th49</a>, <a class="ref" href="setwiseo.html#T50" target="_self" title="SETWISEO:th.50">Th50</a>, <a class="ref" href="setwiseo.html#T51" target="_self" title="SETWISEO:th.51">Th51</a></i>;<br/>
<b>hence </b> <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <span class="p1"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<font color="Maroon" title="c3">f</font> = 
<span class="p1">(<span class="default"><a href="setwiseo.html#K9" title="SETWISEO:func.9">FinUnion</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T27" target="_self" title="SETWISEO:th.27">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#D4" target="_self" title="SETWISEO:def.4">Def4</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/>


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