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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</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">  <a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font> is  <a href="roughs_4.html#V8" title="ROUGHS_4:attr.8">c=-monotone</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">f</font> =  <a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font>;<br/>
<a NAME="E1:47"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">b</font> holds <br/><span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</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="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</font> implies <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1"/><span class="lab"><font color="Green" title="E21">A0</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b</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"> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">b</font></span><br/>

<span class="lab"><font color="Green" title="E22">A1</font></span>: <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> = 
<span class="p1">(<span class="default"><a href="roughs_2.html#K4" title="ROUGHS_2:func.4">UAp</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_5.html#T13" target="_self" onmouseover="rs('roughs_5/T13')" onmouseout="rh()">UApF0</a></span>
<br/>.= 
 <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_2.html#D11" onmouseover="rs('roughs_2/D11')" onmouseout="rh()">ROUGHS_2:def 11</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">b</font> = 
<span class="p1">(<span class="default"><a href="roughs_2.html#K4" title="ROUGHS_2:func.4">UAp</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_5.html#T13" target="_self" onmouseover="rs('roughs_5/T13')" onmouseout="rh()">UApF0</a></span>
<br/>.= 
 <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_2.html#D11" onmouseover="rs('roughs_2/D11')" onmouseout="rh()">ROUGHS_2:def 11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:47_1"/>
<span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:47_1"><span class="lab"><font color="Green" title="E21">A0</font></span></a>, <a class="ref" href="roughs_2.html#T15" onmouseover="rs('roughs_2/T15')" onmouseout="rh()">ROUGHS_2:15</a></span>; <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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:47"/>
 <a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font> is  <a href="roughs_4.html#V8" title="ROUGHS_4:attr.8">c=-monotone</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"> verum</span><br/>


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