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<span class="kw">let </span><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_4.html#K8" title="FUNCT_4:func.8">+~</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_4.html#K8" title="FUNCT_4:func.8">+~</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_4.html#K8" title="FUNCT_4:func.8">+~</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_4.html#K8" title="FUNCT_4:func.8">+~</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_4.html#K8" title="FUNCT_4:func.8">+~</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_4.html#K8" title="FUNCT_4:func.8">+~</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">A</font> = <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">g</font></span>)</span>;<br/>
<a NAME="E1:153"/><span class="lab"><font color="Green" title="E70">A1</font></span>: 
<font color="Maroon" title="c2">g</font> <a href="funct_4.html#K8" title="FUNCT_4:func.8">+~</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span></span>)</span> <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T116" target="_self" onmouseover="rs('funct_7/T116')" onmouseout="rh()">Th115</a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<span class="lab"><font color="Green" title="E71">A2</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span></span>)</span> <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T30" target="_self" onmouseover="rs('funct_7/T30')" onmouseout="rh()">Th29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">g</font></span>)</span>

;<br/>
<a NAME="E3:153"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T17" onmouseover="rs('funct_4/T17')" onmouseout="rh()">FUNCT_4:17</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_4.html#K8" title="FUNCT_4:func.8">+~</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span></span>)</span> <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T116" target="_self" onmouseover="rs('funct_7/T116')" onmouseout="rh()">Th115</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p5">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span></span>)</span> <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p5">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span></span>)</span> <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:153"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="funct_7.html#T9" target="_self" onmouseover="rs('funct_7/T9')" onmouseout="rh()">Th9</a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_4.html#K8" title="FUNCT_4:func.8">+~</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_4.html#K8" title="FUNCT_4:func.8">+~</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:153"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="ref" href="funct_7.html#T116" target="_self" onmouseover="rs('funct_7/T116')" onmouseout="rh()">Th115</a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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