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<span class="kw">let </span><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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</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="b4">g1</font>, <font color="Olive" title="b5">g2</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f2</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b2">f1</font> <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Olive" title="b3">f2</font> holds <br/><font color="Olive" title="b4">g1</font> <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Olive" title="b5">g2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</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">g1</font>, <font color="Olive" title="b4">g2</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b3">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b4">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">f1</font> <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Olive" title="b2">f2</font> holds <br/><font color="Olive" title="b3">g1</font> <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Olive" title="b4">g2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</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" 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">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Maroon" title="c4">f2</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Olive" title="b2">g2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">g1</font>, <font color="Maroon" title="c6">g2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c6">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Maroon" title="c4">f2</font> implies <font color="Maroon" title="c5">g1</font> <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Maroon" title="c6">g2</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:5"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c5">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">and </span><br/><a NAME="E2:5"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c6">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">f1</font> <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Maroon" title="c4">f2</font> or <font color="Maroon" title="c5">g1</font> <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Maroon" title="c6">g2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:5"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
 for <font color="Olive" title="b1">a</font>, <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">D</font> holds  <font color="Maroon" title="c3">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice2.html#D1" onmouseover="rs('lattice2/D1')" onmouseout="rh()">LATTICE2:def 1</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"> <font color="Maroon" title="c5">g1</font> <a href="lattice2.html#R1" title="LATTICE2:pred.1">absorbs</a> <font color="Maroon" title="c6">g2</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice2.html#D1" onmouseover="rs('lattice2/D1')" onmouseout="rh()">LATTICE2:def 1</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"> <font color="Maroon" title="c5">g1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">g2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font></span><br/>

<a NAME="E4:5"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c2">S</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:5"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c2">S</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c5">g1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">g2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>)</span>)</span>) = 
<font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>)</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E4:5"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c3">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>)</span>)</span>)

<br/>.= 
<font color="Maroon" title="c7">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E6">A3</font></span></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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