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<b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>set </b><font color="Maroon">x</font> =  <a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">i1</font>,<font color="Maroon">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">j1</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">j1</font> <a href="hidden.html#NR2" target="_self">&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">E28</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>hence </b> <a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> = 
 <a href="arytm_3.html#K13">{}</a> 
<b>by </b><i><a class="ref" href="arytm_3.html#D10">ARYTM_3:def 10</a></i>
<br/>.= 
<font color="Maroon">i1</font> <a href="real_1.html#K6">/</a> <font color="Maroon">j1</font>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T49">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">E29</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_2"/><b>then </b>
<font color="Maroon">j1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">x1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">x2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">y1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">y2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E5:1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">j1</font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E6:1_1_2"/><i><font color="Green">E36</font></i>: 
1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y2</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x2</font>,<font color="Maroon">y1</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5">XCMPLX_0:def 5</a></i>;<br/><a NAME="E7:1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><a NAME="E8:1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><a NAME="E9:1_1_2"/>
<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><a NAME="E10:1_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">j1</font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><a NAME="E11:1_1_2"/>
 <a href="arytm_0.html#K1">+</a> 0,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#D4">ARYTM_0:def 4</a></i>;<br/><a NAME="E12:1_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="arytm_0.html#K3">opp</a> 0 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>;<br/><a NAME="E13:1_1_2"/>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y2</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x2</font>,<font color="Maroon">y1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><b>then </b><i><font color="Green">E41</font></i>: 1 = 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>
<br/>.= 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>
<b>by </b><i>, <a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
;<br/><b>consider </b><font color="Maroon">x'1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">x'2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">y'1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">y'2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E16:1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x'1</font>,<font color="Maroon">x'2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E17:1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">j1</font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y'1</font>,<font color="Maroon">y'2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E18:1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">i1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x'1</font>,<font color="Maroon">y'1</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x'2</font>,<font color="Maroon">y'2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x'1</font>,<font color="Maroon">y'2</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x'2</font>,<font color="Maroon">y'1</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5">XCMPLX_0:def 5</a></i>;<br/><a NAME="E19:1_1_2"/>
<font color="Maroon">x'2</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><a NAME="E20:1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'1</font>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><a NAME="E21:1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">y'2</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><a NAME="E22:1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">j1</font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y'1</font>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><a NAME="E23:1_1_2"/>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x'1</font>,<font color="Maroon">y'2</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x'2</font>,<font color="Maroon">y'1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/><b>then </b><i><font color="Green">E52</font></i>: <font color="Maroon">i1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K2">"</a> </span>)</span> = 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x'1</font>,<font color="Maroon">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x'2</font>,0</span>)</span></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>
<br/>.= 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x'1</font>,<font color="Maroon">y1</font></span>)</span>,0
<b>by </b><i>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x'1</font>,<font color="Maroon">y1</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
;<br/><a NAME="E25:1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x'1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i/>;<br/><a NAME="E26:1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">x'1</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/><a NAME="E27:1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/><b>consider </b><font color="Maroon">yy</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E30:1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">xx</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">yy</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K14">one</a> 
 <b>by </b><i>, , <a class="ref" href="arytm_3.html#T60">ARYTM_3:60</a></i>;<br/><a NAME="E31:1_1_2"/>
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">yy</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon">xx1</font> = <font color="Maroon">xx</font>, <font color="Maroon">yy1</font> = <font color="Maroon">yy</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="ref" href="arytm_2.html#T1">ARYTM_2:1</a></i>;<br/><a NAME="E33:1_1_2"/><i><font color="Green">E60</font></i>: 
 ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">xx1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Maroon">yy1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp; <font color="Maroon">xx1</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">yy1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">y'</font> )
 <b>by </b><i><a class="ref" href="arytm_2.html#T22">ARYTM_2:22</a></i>;<br/><a NAME="E34:1_1_2"/>
( <font color="Maroon">xx1</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  &amp; <font color="Maroon">yy1</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon">xx1</font> = <font color="Maroon">xx1</font>, <font color="Maroon">yy1</font> = <font color="Maroon">yy1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="arytm_0.html#T1">ARYTM_0:1</a></i>;<br/><a NAME="E36:1_1_2"/>
 ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">xx1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Maroon">yy1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp;  <a href="arytm_0.html#K2">*</a> <font color="Maroon">xx1</font>,<font color="Maroon">yy1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Olive">y'</font> )
 <b>by </b><i><a class="ref" href="arytm_0.html#D3">ARYTM_0:def 3</a></i>;<br/><a NAME="E37:1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">yy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i>, , , , , <a class="ref" href="arytm_0.html#T20">ARYTM_0:20</a></i>;<br/><a NAME="E38:1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><b>then consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E40:1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x'1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E41:1_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E42:1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">i1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="ref" href="arytm_0.html#D3">ARYTM_0:def 3</a>, <a class="ref" href="arytm_2.html#T1">ARYTM_2:1</a></i>;<br/><b>consider </b><font color="Maroon">x''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">y''</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E44:1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font>
 <b>and </b><br/><a NAME="E45:1_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y''</font>
 <b>and </b><br/><a NAME="E46:1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">i1</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">y''</font>
 <b>by </b><i>, , , , , <a class="ref" href="arytm_2.html#T22">ARYTM_2:22</a>, <a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/><a NAME="E47:1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> =  <a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font>, <font color="Maroon">b</font> = <font color="Maroon">j1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/><a NAME="E49:1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">x1'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> , <font color="Maroon">y1'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E51:1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1'</font>
 <b>and </b><br/><a NAME="E52:1_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1'</font>
 <b>and </b><br/><a NAME="E53:1_1_2"/><i><font color="Green">E78</font></i>: 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y1'</font>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#D3">ARYTM_0:def 3</a>, <a class="ref" href="arytm_2.html#T1">ARYTM_2:1</a>, <a class="ref" href="arytm_2.html#T2">ARYTM_2:2</a></i>;<br/><a NAME="E54:1_1_2"/><i><font color="Green">E79</font></i>: 
 ex <font color="Olive">x1''</font>, <font color="Olive">y1''</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">x1'</font> <a href="hidden.html#R1">=</a> <font color="Olive">x1''</font> &amp; <font color="Maroon">y1'</font> <a href="hidden.html#R1">=</a> <font color="Olive">y1''</font> &amp; <font color="Maroon">x1'</font> <a href="arytm_2.html#K8">*'</a> <font color="Maroon">y1'</font> <a href="hidden.html#R1">=</a> <font color="Olive">x1''</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">y1''</font> )
 <b>by </b><i>, , , , , <a class="ref" href="arytm_2.html#T22">ARYTM_2:22</a>, <a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/><font color="Maroon">a</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">j1</font>,<a href="arytm_3.html#K14">one</a> </span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T46">ARYTM_3:46</a></i>
<br/>.= 
 <a href="arytm_3.html#NK11" target="_self">quotient</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="arytm_3.html#K3">*^</a> <font color="Maroon">j1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="arytm_3.html#K3">*^</a> <a href="arytm_3.html#K14">one</a> </span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T55">ARYTM_3:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">i1</font>,<a href="arytm_3.html#K14">one</a> </span>)</span> <a href="arytm_3.html#K12">*'</a> <span class="p1">(<span class="default"><a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">j1</font>,<font color="Maroon">j1</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T55">ARYTM_3:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">i1</font>,<a href="arytm_3.html#K14">one</a> </span>)</span> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T47">ARYTM_3:47</a></i>
<br/>.= 
 <a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">i1</font>,<a href="arytm_3.html#K14">one</a> 
<b>by </b><i><a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>
<br/>.= 
<font color="Maroon">i1</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T46">ARYTM_3:46</a></i>
<br/>.= 
<font color="Maroon">x''</font> <a href="arytm_3.html#K12">*'</a> <a href="arytm_3.html#K14">one</a> 
<b>by </b><i>, , <a class="txt" href="rat_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="arytm_3.html#T59">ARYTM_3:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x''</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">y''</font></span>)</span> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">b</font>
<b>by </b><i>, , , , , , , <a class="txt" href="rat_1.html#E4:1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="arytm_3.html#T58">ARYTM_3:58</a></i>
;<br/><b>hence </b><a NAME="E56:1_1_2"/>
 <a href="arytm_3.html#NK11" target="_self">quotient</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="real_1.html#K6">/</a> <font color="Maroon">j1</font>
 <b>by </b><i>, <a class="txt" href="rat_1.html#E1:1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="arytm_3.html#T62">ARYTM_3:62</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
