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<b>let </b><font color="Maroon" title="c1">i1</font>, <font color="Maroon" title="c2">j1</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">x</font> =  <a href="arytm_3.html#NK11" title="ARYTM_3:NK.11">quotient</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">j1</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1"/>
( <font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  or <font color="Maroon" title="c2">j1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><div class="add"><b>hence </b> <a href="arytm_3.html#NK11" title="ARYTM_3:NK.11">quotient</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">j1</font> = 
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
<b>by </b><i><a class="ref" href="arytm_3.html#D10" title="ARYTM_3:def.10">ARYTM_3:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c1">i1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">j1</font>
<b>by </b><i><a class="txt" href="rat_1.html#E1:1_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T49" title="XCMPLX_1:th.49">XCMPLX_1:49</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2"><b>suppose </b></a><a NAME="E1:1_1_2"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c2">j1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_2"/><b>then </b>
<font color="Maroon" title="c2">j1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font>, <font color="Maroon" title="c6">y1</font>, <font color="Maroon" title="c7">y2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:1_1_2"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E5:1_1_2"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon" title="c2">j1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E6:1_1_2"/><i><font color="Green" title="E6">A5</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5" title="XCMPLX_0:def.5">XCMPLX_0:def 5</a></i>;<br/><a NAME="E7:1_1_2"/><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E4:1_1_2"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="arytm_0.html#T26" title="ARYTM_0:th.26">ARYTM_0:26</a></i>;<br/><a NAME="E8:1_1_2"/><b>then </b><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E4:1_1_2"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></i>;<br/><a NAME="E9:1_1_2"/>
<font color="Maroon" title="c7">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E5:1_1_2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="arytm_0.html#T26" title="ARYTM_0:th.26">ARYTM_0:26</a></i>;<br/><a NAME="E10:1_1_2"/><b>then </b><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon" title="c2">j1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E5:1_1_2"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></i>;<br/><a NAME="E11:1_1_2"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> 0,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></i>;<br/><a NAME="E12:1_1_2"/><b>then </b><i><font color="Green" title="E10">A9</font></i>: 
 <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></i>;<br/><a NAME="E13:1_1_2"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E6:1_1_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="arytm_0.html#T26" title="ARYTM_0:th.26">ARYTM_0:26</a></i>;<br/><b>then </b><i><font color="Green" title="E11">A10</font></i>: 1 = 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rat_1.html#E6:1_1_2"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="rat_1.html#E7:1_1_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>
<b>by </b><i><a class="txt" href="rat_1.html#E12:1_1_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></i>
;<br/><b>consider </b><font color="Maroon" title="c8">x'1</font>, <font color="Maroon" title="c9">x'2</font>, <font color="Maroon" title="c10">y'1</font>, <font color="Maroon" title="c11">y'2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E16:1_1_2"/><i><font color="Green" title="E12">A11</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c8">x'1</font>,<font color="Maroon" title="c9">x'2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E17:1_1_2"/><i><font color="Green" title="E13">A12</font></i>: 
<font color="Maroon" title="c2">j1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c10">y'1</font>,<font color="Maroon" title="c11">y'2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E18:1_1_2"/><i><font color="Green" title="E14">A13</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c8">x'1</font>,<font color="Maroon" title="c10">y'1</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c9">x'2</font>,<font color="Maroon" title="c11">y'2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c8">x'1</font>,<font color="Maroon" title="c11">y'2</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c9">x'2</font>,<font color="Maroon" title="c10">y'1</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5" title="XCMPLX_0:def.5">XCMPLX_0:def 5</a></i>;<br/><a NAME="E19:1_1_2"/>
<font color="Maroon" title="c9">x'2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E16:1_1_2"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="arytm_0.html#T26" title="ARYTM_0:th.26">ARYTM_0:26</a></i>;<br/><a NAME="E20:1_1_2"/><b>then </b><i><font color="Green" title="E15">A14</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x'1</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E16:1_1_2"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></i>;<br/><a NAME="E21:1_1_2"/><i><font color="Green" title="E16">A15</font></i>: 
<font color="Maroon" title="c11">y'2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E17:1_1_2"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="arytm_0.html#T26" title="ARYTM_0:th.26">ARYTM_0:26</a></i>;<br/><a NAME="E22:1_1_2"/><b>then </b><i><font color="Green" title="E17">A16</font></i>: 
<font color="Maroon" title="c2">j1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'1</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E17:1_1_2"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></i>;<br/><a NAME="E23:1_1_2"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c8">x'1</font>,<font color="Maroon" title="c11">y'2</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c9">x'2</font>,<font color="Maroon" title="c10">y'1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="rat_1.html#E18:1_1_2"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="arytm_0.html#T26" title="ARYTM_0:th.26">ARYTM_0:26</a></i>;<br/><b>then </b><i><font color="Green" title="E18">A17</font></i>: <font color="Maroon" title="c1">i1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> = 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c8">x'1</font>,<font color="Maroon" title="c6">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c9">x'2</font>,0</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rat_1.html#E10:1_1_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="rat_1.html#E18:1_1_2"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="rat_1.html#E21:1_1_2"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="rat_1.html#E22:1_1_2"><i><font color="Green" title="E17">A16</font></i></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c8">x'1</font>,<font color="Maroon" title="c6">y1</font></span>)</span>,0
<b>by </b><i><a class="txt" href="rat_1.html#E12:1_1_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c8">x'1</font>,<font color="Maroon" title="c6">y1</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></i>
;<br/><a NAME="E25:1_1_2"/><i><font color="Green" title="E19">A18</font></i>: 
<font color="Maroon" title="c8">x'1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E20:1_1_2"><i><font color="Green" title="E15">A14</font></i></a></i>;<br/><a NAME="E26:1_1_2"/><b>then </b><i><font color="Green" title="E20">A19</font></i>: 
<font color="Maroon" title="c8">x'1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E27:1_1_2"/>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E8:1_1_2"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">xx</font> = <font color="Maroon" title="c4">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  <b>by </b><i><a class="txt" href="rat_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c13">yy</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E30:1_1_2"/><i><font color="Green" title="E21">A20</font></i>: 
<font color="Maroon" title="c12">xx</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c13">yy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E1:1_1_2"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="rat_1.html#E8:1_1_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="arytm_3.html#T60" title="ARYTM_3:th.60">ARYTM_3:60</a></i>;<br/><a NAME="E31:1_1_2"/>
( <font color="Maroon" title="c12">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  &amp; <font color="Maroon" title="c13">yy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c14">xx1</font> = <font color="Maroon" title="c12">xx</font>, <font color="Maroon" title="c15">yy1</font> = <font color="Maroon" title="c13">yy</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  <b>by </b><i><a class="ref" href="arytm_2.html#T1" title="ARYTM_2:th.1">ARYTM_2:1</a></i>;<br/><a NAME="E33:1_1_2"/><i><font color="Green" title="E22">A21</font></i>: 
 ex <font color="Olive" title="b1">x'</font>, <font color="Olive" title="b2">y'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  st <br/>( <font color="Maroon" title="c14">xx1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Maroon" title="c15">yy1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y'</font> &amp; <font color="Maroon" title="c14">xx1</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c15">yy1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">y'</font> )
 <b>by </b><i><a class="ref" href="arytm_2.html#T22" title="ARYTM_2:th.22">ARYTM_2:22</a></i>;<br/><a NAME="E34:1_1_2"/>
( <font color="Maroon" title="c14">xx1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  &amp; <font color="Maroon" title="c15">yy1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c16">xx1</font> = <font color="Maroon" title="c14">xx1</font>, <font color="Maroon" title="c17">yy1</font> = <font color="Maroon" title="c15">yy1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="arytm_0.html#T1" title="ARYTM_0:th.1">ARYTM_0:1</a></i>;<br/><a NAME="E36:1_1_2"/>
 ex <font color="Olive" title="b1">x'</font>, <font color="Olive" title="b2">y'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a>  st <br/>( <font color="Maroon" title="c16">xx1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Maroon" title="c17">yy1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y'</font> &amp;  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c16">xx1</font>,<font color="Maroon" title="c17">yy1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Olive" title="b2">y'</font> )
 <b>by </b><i><a class="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a></i>;<br/><a NAME="E37:1_1_2"/><b>then </b><i><font color="Green" title="E23">A22</font></i>: 
<font color="Maroon" title="c13">yy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E1:1_1_2"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="rat_1.html#E8:1_1_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="rat_1.html#E14:1_1_2"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="rat_1.html#E30:1_1_2"><i><font color="Green" title="E21">A20</font></i></a>, <a class="txt" href="rat_1.html#E33:1_1_2"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="arytm_0.html#T20" title="ARYTM_0:th.20">ARYTM_0:20</a></i>;<br/><a NAME="E38:1_1_2"/><b>then </b><i><font color="Green" title="E24">A23</font></i>: 
<font color="Maroon" title="c6">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c18">x'</font>, <font color="Maroon" title="c19">y'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E40:1_1_2"/><i><font color="Green" title="E25">A24</font></i>: 
<font color="Maroon" title="c8">x'1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x'</font>
 <b>and </b><br/><a NAME="E41:1_1_2"/><i><font color="Green" title="E26">A25</font></i>: 
<font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y'</font>
 <b>and </b><br/><a NAME="E42:1_1_2"/><i><font color="Green" title="E27">A26</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c19">y'</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E24:1_1_2"><i><font color="Green" title="E18">A17</font></i></a>, <a class="txt" href="rat_1.html#E26:1_1_2"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a>, <a class="ref" href="arytm_2.html#T1" title="ARYTM_2:th.1">ARYTM_2:1</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">x''</font>, <font color="Maroon" title="c21">y''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E44:1_1_2"/><i><font color="Green" title="E28">A27</font></i>: 
<font color="Maroon" title="c18">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x''</font>
 <b>and </b><br/><a NAME="E45:1_1_2"/><i><font color="Green" title="E29">A28</font></i>: 
<font color="Maroon" title="c19">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y''</font>
 <b>and </b><br/><a NAME="E46:1_1_2"/><i><font color="Green" title="E30">A29</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x''</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">y''</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E25:1_1_2"><i><font color="Green" title="E19">A18</font></i></a>, <a class="txt" href="rat_1.html#E37:1_1_2"><i><font color="Green" title="E23">A22</font></i></a>, <a class="txt" href="rat_1.html#E40:1_1_2"><i><font color="Green" title="E25">A24</font></i></a>, <a class="txt" href="rat_1.html#E41:1_1_2"><i><font color="Green" title="E26">A25</font></i></a>, <a class="txt" href="rat_1.html#E42:1_1_2"><i><font color="Green" title="E27">A26</font></i></a>, <a class="ref" href="arytm_2.html#T22" title="ARYTM_2:th.22">ARYTM_2:22</a>, <a class="txt" href="rat_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E47:1_1_2"/><i><font color="Green" title="E31">A30</font></i>: 
<font color="Maroon" title="c2">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c22">a</font> =  <a href="arytm_3.html#NK11" title="ARYTM_3:NK.11">quotient</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">j1</font>, <font color="Maroon" title="c23">b</font> = <font color="Maroon" title="c2">j1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  <b>by </b><i><a class="txt" href="rat_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E49:1_1_2"/>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="txt" href="rat_1.html#E8:1_1_2"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">x1'</font>, <font color="Maroon" title="c25">y1'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E51:1_1_2"/><i><font color="Green" title="E32">A31</font></i>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x1'</font>
 <b>and </b><br/><a NAME="E52:1_1_2"/><i><font color="Green" title="E33">A32</font></i>: 
<font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">y1'</font>
 <b>and </b><br/><a NAME="E53:1_1_2"/><i><font color="Green" title="E34">A33</font></i>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x1'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c25">y1'</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E38:1_1_2"><i><font color="Green" title="E24">A23</font></i></a>, <a class="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a>, <a class="ref" href="arytm_2.html#T1" title="ARYTM_2:th.1">ARYTM_2:1</a>, <a class="ref" href="arytm_2.html#T2" title="ARYTM_2:th.2">ARYTM_2:2</a></i>;<br/><a NAME="E54:1_1_2"/><i><font color="Green" title="E35">A34</font></i>: 
 ex <font color="Olive" title="b1">x1''</font>, <font color="Olive" title="b2">y1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  st <br/>( <font color="Maroon" title="c24">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1''</font> &amp; <font color="Maroon" title="c25">y1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y1''</font> &amp; <font color="Maroon" title="c24">x1'</font> <a href="arytm_2.html#K8" title="ARYTM_2:func.8">*'</a> <font color="Maroon" title="c25">y1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1''</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Olive" title="b2">y1''</font> )
 <b>by </b><i><a class="txt" href="rat_1.html#E8:1_1_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="rat_1.html#E37:1_1_2"><i><font color="Green" title="E23">A22</font></i></a>, <a class="txt" href="rat_1.html#E47:1_1_2"><i><font color="Green" title="E31">A30</font></i></a>, <a class="txt" href="rat_1.html#E51:1_1_2"><i><font color="Green" title="E32">A31</font></i></a>, <a class="txt" href="rat_1.html#E52:1_1_2"><i><font color="Green" title="E33">A32</font></i></a>, <a class="ref" href="arytm_2.html#T22" title="ARYTM_2:th.22">ARYTM_2:22</a>, <a class="txt" href="rat_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><font color="Maroon" title="c22">a</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c23">b</font> = 
<span class="p1">(<span class="default"><a href="arytm_3.html#NK11" title="ARYTM_3:NK.11">quotient</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">j1</font></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><a href="arytm_3.html#NK11" title="ARYTM_3:NK.11">quotient</a> <font color="Maroon" title="c2">j1</font>,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T46" title="ARYTM_3:th.46">ARYTM_3:46</a></i>
<br/>.= 
 <a href="arytm_3.html#NK11" title="ARYTM_3:NK.11">quotient</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <font color="Maroon" title="c2">j1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="arytm_3.html#K3" title="ARYTM_3:func.3">*^</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T55" title="ARYTM_3:th.55">ARYTM_3:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#NK11" title="ARYTM_3:NK.11">quotient</a> <font color="Maroon" title="c1">i1</font>,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <span class="p1">(<span class="default"><a href="arytm_3.html#NK11" title="ARYTM_3:NK.11">quotient</a> <font color="Maroon" title="c2">j1</font>,<font color="Maroon" title="c2">j1</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T55" title="ARYTM_3:th.55">ARYTM_3:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#NK11" title="ARYTM_3:NK.11">quotient</a> <font color="Maroon" title="c1">i1</font>,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> </span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
<b>by </b><i><a class="txt" href="rat_1.html#E1:1_1_2"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="arytm_3.html#T47" title="ARYTM_3:th.47">ARYTM_3:47</a></i>
<br/>.= 
 <a href="arytm_3.html#NK11" title="ARYTM_3:NK.11">quotient</a> <font color="Maroon" title="c1">i1</font>,<a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
<b>by </b><i><a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></i>
<br/>.= 
<font color="Maroon" title="c1">i1</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T46" title="ARYTM_3:th.46">ARYTM_3:46</a></i>
<br/>.= 
<font color="Maroon" title="c20">x''</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <a href="arytm_3.html#K14" title="ARYTM_3:func.14">one</a> 
<b>by </b><i><a class="txt" href="rat_1.html#E20:1_1_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="rat_1.html#E40:1_1_2"><i><font color="Green" title="E25">A24</font></i></a>, <a class="txt" href="rat_1.html#E44:1_1_2"><i><font color="Green" title="E28">A27</font></i></a>, <a class="ref" href="arytm_3.html#T59" title="ARYTM_3:th.59">ARYTM_3:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">x''</font> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c21">y''</font></span>)</span> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">*'</a> <font color="Maroon" title="c23">b</font>
<b>by </b><i><a class="txt" href="rat_1.html#E8:1_1_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="rat_1.html#E14:1_1_2"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="rat_1.html#E41:1_1_2"><i><font color="Green" title="E26">A25</font></i></a>, <a class="txt" href="rat_1.html#E45:1_1_2"><i><font color="Green" title="E29">A28</font></i></a>, <a class="txt" href="rat_1.html#E51:1_1_2"><i><font color="Green" title="E32">A31</font></i></a>, <a class="txt" href="rat_1.html#E52:1_1_2"><i><font color="Green" title="E33">A32</font></i></a>, <a class="txt" href="rat_1.html#E53:1_1_2"><i><font color="Green" title="E34">A33</font></i></a>, <a class="txt" href="rat_1.html#E54:1_1_2"><i><font color="Green" title="E35">A34</font></i></a>, <a class="ref" href="arytm_3.html#T58" title="ARYTM_3:th.58">ARYTM_3:58</a></i>
;<br/><b>hence </b><a NAME="E56:1_1_2"/>
 <a href="arytm_3.html#NK11" title="ARYTM_3:NK.11">quotient</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">j1</font>
 <b>by </b><i><a class="txt" href="rat_1.html#E1:1_1_2"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="rat_1.html#E46:1_1_2"><i><font color="Green" title="E30">A29</font></i></a>, <a class="ref" href="arytm_3.html#T62" title="ARYTM_3:th.62">ARYTM_3:62</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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