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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="surreal0.html#NM1" title="SURREAL0:NM.1">Surreal</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="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c1">x</font> is  <a href="surrealn.html#V2" title="SURREALN:attr.2">*real</a>  implies <font color="Maroon" title="c2">y</font> is  <a href="surrealn.html#V2" title="SURREALN:attr.2">*real</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:106"/><span class="lab"><font color="Green" title="E87">A1</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c1">x</font> is  <a href="surrealn.html#V2" title="SURREALN:attr.2">*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"> <font color="Maroon" title="c2">y</font> is  <a href="surrealn.html#V2" title="SURREALN:attr.2">*real</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:106"/><span class="lab"><font color="Green" title="E88">A2</font></span>: 
( <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c1">x</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> )
 ;<br/>
<a NAME="E4:106"/><span class="lab"><font color="Green" title="E89">A3</font></span>: 
( <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c2">y</font> <a href="surrealo.html#NR3" title="SURREALO:NR.3">&lt;</a> <a href="surrealn.html#K1" title="SURREALN:func.1">uInt</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:106"><span class="lab"><font color="Green" title="E88">A2</font></span></a>, <a class="txt" href="#E1:106"><span class="lab"><font color="Green" title="E87">A1</font></span></a>, <a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<a NAME="E5:106"/>
( <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font> &amp;  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c2">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106"><span class="lab"><font color="Green" title="E87">A1</font></span></a>, <a class="txt" href="surrealn.html#E88"><span class="lab"><font color="Green" title="E83">Lm20</font></span></a></span>;<br/>
<a NAME="E6:106"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c2">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="surrealo.html#T4" onmouseover="rs('surrealo/T4')" onmouseout="rh()">SURREALO:4</a></span>;<br/>
<a NAME="E7:106"/><span class="kw">then </span>
<font color="Maroon" title="c2">y</font> <a href="surrealo.html#R2" title="SURREALO:pred.2">==</a>  <a href="surrealn.html#K7" title="SURREALN:func.7">real_qua</a> <font color="Maroon" title="c2">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106"><span class="lab"><font color="Green" title="E87">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="E8:106"/>
<font color="Maroon" title="c2">y</font> is  <a href="surrealn.html#V2" title="SURREALN:attr.2">*real</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:106"><span class="lab"><font color="Green" title="E89">A3</font></span></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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