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<span class="kw">let </span><font color="Maroon" title="c1">r1</font>, <font color="Maroon" title="c2">r2</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> iff <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r2</font> )</span><br/>



<a NAME="E1:82"/>
( <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r1</font></span>)</span> &amp; <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surrealo.html#K6" title="SURREALO:func.6">Unique_No</a> <span class="p1">(<span class="default"><a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#D7" target="_self" onmouseover="rs('surrealn/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E2:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A1</font></span>: 
( <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> &amp; <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#D10" onmouseover="rs('surrealo/D10')" onmouseout="rh()">SURREALO:def 10</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:82"/>
( <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> implies <font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r2</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="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r2</font> implies <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r2</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="82_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:82_1"/>
<a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r2</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="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r2</font></span><br/>

<a NAME="E2:82_1"/><span class="kw">then </span>
<a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<a NAME="E3:82_1"/><span class="kw">then </span>
<a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:82_1"/>
<font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="surrealn.html#T50" target="_self" onmouseover="rs('surrealn/T50')" onmouseout="rh()">Th50</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">assume </span><a NAME="E4:82"/>
<font color="Maroon" title="c1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r2</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"> <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r2</font></span><br/>

<a NAME="E5:82"/><span class="kw">then </span>
<a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K5" title="SURREALN:func.5">sReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a>, <a class="ref" href="surrealn.html#T50" target="_self" onmouseover="rs('surrealn/T50')" onmouseout="rh()">Th50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:82"/>
<a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">r1</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K6" title="SURREALN:func.6">uReal</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</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/>


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