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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_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="E1:14_1"/>
( <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>  or <font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="14_1_1"><span class="kw">suppose </span></a><a NAME="E1:14_1_1"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<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> 
 ; <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="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><a NAME="E2:14_1_1"/><span class="kw">then </span>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="prgcor_1.html#D2" target="_self" onmouseover="rs('prgcor_1/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:14_1_1"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_1"><span class="lab"><font color="Green" title="E13">A1</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2"><span class="kw">suppose </span></a><a NAME="E1:14_1_2"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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">  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><span class="kw">now </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"> ( ( <font color="Maroon" title="c1">n</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>  &amp;  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font> ) or ( <font color="Maroon" title="c1">n</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>  &amp;  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_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="E1:14_1_2_1_1"/>
( <font color="Maroon" title="c1">n</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="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="14_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:14_1_2_1_1_1"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c1">n</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">  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_1_1_1"><span class="kw">now </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"> ( ( <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font> ) or ( <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>  &amp;  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1_1_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="E1:14_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2"><span class="lab"><font color="Green" title="E13">A2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:14_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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">  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><span class="kw">hence </span> <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) = 
 <a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="prgcor_1.html#D2" target="_self" onmouseover="rs('prgcor_1/D2')" onmouseout="rh()">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="prgcor_1.html#T10" target="_self" onmouseover="rs('prgcor_1/T10')" onmouseout="rh()">Th10</a></span>
;<br/> <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="14_1_2_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:14_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<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">  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_1_1_1_1_2_1"><span class="kw">now </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"> ( ( <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font> ) or ( <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1_1_1_1_2_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="E1:14_1_2_1_1_1_1_1_2_1_1"/>
( <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> or <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1_1_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:14_1_2_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</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="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">m2</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c4">n2</font> = <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><a NAME="E3:14_1_2_1_1_1_1_1_2_1_1_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="xreal_1.html#T58" onmouseover="rs('xreal_1/T58')" onmouseout="rh()">XREAL_1:58</a></span>;<br/><a NAME="E4:14_1_2_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A7</font></span>: 
 <a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prgcor_1.html#T9" target="_self" onmouseover="rs('prgcor_1/T9')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E5:14_1_2_1_1_1_1_1_2_1_1_1"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="prgcor_1.html#D2" target="_self" onmouseover="rs('prgcor_1/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:14_1_2_1_1_1_1_1_2_1_1_1"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E4:14_1_2_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="prgcor_1.html#T11" target="_self" onmouseover="rs('prgcor_1/T11')" onmouseout="rh()">Th11</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="14_1_2_1_1_1_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:14_1_2_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</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="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">m2</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c4">n2</font> = <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><a NAME="E3:14_1_2_1_1_1_1_1_2_1_1_2"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="xreal_1.html#T58" onmouseover="rs('xreal_1/T58')" onmouseout="rh()">XREAL_1:58</a></span>;<br/><a NAME="E4:14_1_2_1_1_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A9</font></span>: 
 <a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prgcor_1.html#T9" target="_self" onmouseover="rs('prgcor_1/T9')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E5:14_1_2_1_1_1_1_1_2_1_1_2"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<font color="Maroon" title="c1">n</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="prgcor_1.html#D2" target="_self" onmouseover="rs('prgcor_1/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:14_1_2_1_1_1_1_1_2_1_1_2"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E4:14_1_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="prgcor_1.html#T11" target="_self" onmouseover="rs('prgcor_1/T11')" onmouseout="rh()">Th11</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></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:14_1_2_1_1_1_1_1_2"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:14_1_2_1_1_1"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:14_1_2_1_1_2"/><span class="lab"><font color="Green" title="E14">A10</font></span>: 
<font color="Maroon" title="c1">n</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">  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_1_2_1"><span class="kw">now </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"> ( ( <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>  &amp;  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font> ) or ( <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1_2_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="E1:14_1_2_1_1_2_1_1"/>
( <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>  or <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2"><span class="lab"><font color="Green" title="E13">A2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:14_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E15">A11</font></span>: 
<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">  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><a NAME="E2:14_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A12</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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_1.html#T58" onmouseover="rs('xreal_1/T58')" onmouseout="rh()">XREAL_1:58</a></span>;<br/><span class="lab"><font color="Green" title="E17">A13</font></span>: <font color="Maroon" title="c1">n</font> <a href="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font> = 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">m</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D9" onmouseover="rs('int_1/D9')" onmouseout="rh()">INT_1:def 9</a></span>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T191" onmouseover="rs('xcmplx_1/T191')" onmouseout="rh()">XCMPLX_1:191</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="int_1.html#K4" title="INT_1:func.4">div</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D9" onmouseover="rs('int_1/D9')" onmouseout="rh()">INT_1:def 9</a></span>
;<br/> <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) = 
 <a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_2"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="ref" href="prgcor_1.html#D2" target="_self" onmouseover="rs('prgcor_1/D2')" onmouseout="rh()">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="int_1.html#K4" title="INT_1:func.4">div</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_2"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E2:14_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="ref" href="prgcor_1.html#T10" target="_self" onmouseover="rs('prgcor_1/T10')" onmouseout="rh()">Th10</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:14_1_2_1_1_2_1_1_1"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E17">A13</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:14_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E15">A14</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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">  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_1_2_1_1_2_1"><span class="kw">now </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"> ( ( <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font> ) or ( <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1_2_1_1_2_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="E1:14_1_2_1_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2_1_1_2_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:14_1_2_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E16">A15</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</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="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">m2</font> = <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c4">n2</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_2"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><a NAME="E3:14_1_2_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E17">A16</font></span>: 
 <a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="ref" href="prgcor_1.html#T9" target="_self" onmouseover="rs('prgcor_1/T9')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E4:14_1_2_1_1_2_1_1_2_1_1_1"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">m</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_2"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="ref" href="prgcor_1.html#D2" target="_self" onmouseover="rs('prgcor_1/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:14_1_2_1_1_2_1_1_2_1_1_1"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_2"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E3:14_1_2_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="ref" href="prgcor_1.html#T11" target="_self" onmouseover="rs('prgcor_1/T11')" onmouseout="rh()">Th11</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="14_1_2_1_1_2_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:14_1_2_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E16">A17</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</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="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">m2</font> = <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c4">n2</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_2"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><a NAME="E3:14_1_2_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E17">A18</font></span>: 
 <a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="ref" href="prgcor_1.html#T9" target="_self" onmouseover="rs('prgcor_1/T9')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E4:14_1_2_1_1_2_1_1_2_1_1_2"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="prgcor_1.html#K1" title="PRGCOR_1:func.1">idiv1_prg</a> (<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">m</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_2"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="ref" href="prgcor_1.html#D2" target="_self" onmouseover="rs('prgcor_1/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:14_1_2_1_1_2_1_1_2_1_1_2"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_1_1_2"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E1:14_1_2_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="txt" href="#E3:14_1_2_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="ref" href="prgcor_1.html#T11" target="_self" onmouseover="rs('prgcor_1/T11')" onmouseout="rh()">Th11</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></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:14_1_2_1_1_2_1_1_2"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:14_1_2_1_1_2"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:14_1_2"/>
 <a href="prgcor_1.html#K2" title="PRGCOR_1:func.2">idiv_prg</a> (<font color="Maroon" title="c1">n</font>,<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="int_1.html#K4" title="INT_1:func.4">div</a> <font color="Maroon" title="c2">m</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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