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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">x</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">y</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_1_1_1"/>
( <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E4:2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E5:2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:2_1_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E8:2_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_2_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:2_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E5:2_1_1_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:2_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>


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