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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">m</font></span><br/>

<a NAME="E1:1"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">m</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1"/>
( <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><a NAME="E2:1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_2"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><a NAME="E2:1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E1:1_1_2"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</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></div><span class="kw">end;</span></div>

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