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



<b>assume </b><a NAME="E1:68"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<b>consider </b><font color="Maroon">m</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">n</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:68"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E4:68"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">k</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> 1;<br/>
<a NAME="E5:68"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#T57">INT_1:57</a></i>;<br/>
<a NAME="E6:68"/><b>then </b>
( <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span> &amp; <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T31">XXREAL_0:31</a></i>;<br/>
<a NAME="E7:68"/><b>then </b>
( <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font> &amp; <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>;<br/>
<a NAME="E8:68"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K2">[/</a><span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K2">[/</a><span class="default"><span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="real_2.html#T177">REAL_2:177</a></i>;<br/>
<a NAME="E9:68"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="int_1.html#K2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="int_1.html#K2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">e</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="int_1.html#K2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>, <font color="Maroon">d</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="int_1.html#K2">[/</a><span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E11:68"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E12:68"/>
 ex <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">c</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Olive">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a></i>;<br/>


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