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<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <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> ; <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="c5">x</font> <a href="number14.html#R1" title="NUMBER14:pred.1">solves_CRT</a> <font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c3">n1</font>,<font color="Maroon" title="c2">a2</font>,<font color="Maroon" title="c4">n2</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n2</font> &amp; <font color="Maroon" title="c6">y</font> <a href="number14.html#R1" title="NUMBER14:pred.1">solves_CRT</a> <font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c3">n1</font>,<font color="Maroon" title="c2">a2</font>,<font color="Maroon" title="c4">n2</font> &amp; <font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n2</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_1_2"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="number14.html#R1" title="NUMBER14:pred.1">solves_CRT</a> <font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c3">n1</font>,<font color="Maroon" title="c2">a2</font>,<font color="Maroon" title="c4">n2</font>
 <span class="kw">and </span><br/><a NAME="E2:27_1_2"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n2</font>
 <span class="kw">and </span><br/><a NAME="E3:27_1_2"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="number14.html#R1" title="NUMBER14:pred.1">solves_CRT</a> <font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c3">n1</font>,<font color="Maroon" title="c2">a2</font>,<font color="Maroon" title="c4">n2</font>
 <span class="kw">and </span><br/><a NAME="E4:27_1_2"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n2</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"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font></span><br/>

<a NAME="E5:27_1_2"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c3">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T11" onmouseover="rs('int_1/T11')" onmouseout="rh()">INT_1:11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:27_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</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"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2_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="E1:27_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">y</font> or <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">y</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:27_1_2_1_1"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">y</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"> contradiction</span><br/><div class="add"><a NAME="E2:27_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A11</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">y</font> is    <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="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><a NAME="E3:27_1_2_1_1"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c1">a1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a1</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c3">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E3:27_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="int_1.html#T17" onmouseover="rs('int_1/T17')" onmouseout="rh()">INT_1:17</a></span>;<br/><a NAME="E4:27_1_2_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c2">a2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">a2</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c4">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E3:27_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="int_1.html#T17" onmouseover="rs('int_1/T17')" onmouseout="rh()">INT_1:17</a></span>;<br/><a NAME="E5:27_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c3">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:27_1_2_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E3:27_1_2_1_1"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="pepin.html#T4" onmouseover="rs('pepin/T4')" onmouseout="rh()">PEPIN:4</a></span>;<br/><span class="kw">hence </span><a NAME="E6:27_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E4:27_1_2"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E5:27_1_2"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="txt" href="#E1:27_1_2_1_1"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="nat_6.html#T14" onmouseover="rs('nat_6/T14')" onmouseout="rh()">NAT_6:14</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="27_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:27_1_2_1_2"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">y</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"> contradiction</span><br/><div class="add"><a NAME="E2:27_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A14</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">x</font> is    <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="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><a NAME="E3:27_1_2_1_2"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c1">a1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a1</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c3">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E3:27_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="int_1.html#T17" onmouseover="rs('int_1/T17')" onmouseout="rh()">INT_1:17</a></span>;<br/><a NAME="E4:27_1_2_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">a2</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c4">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E3:27_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="int_1.html#T17" onmouseover="rs('int_1/T17')" onmouseout="rh()">INT_1:17</a></span>;<br/><a NAME="E5:27_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c3">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:27_1_2_1_2"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E3:27_1_2_1_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="pepin.html#T4" onmouseover="rs('pepin/T4')" onmouseout="rh()">PEPIN:4</a></span>;<br/><a NAME="E6:27_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c3">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T14" onmouseover="rs('int_1/T14')" onmouseout="rh()">INT_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E7:27_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E4:27_1_2"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E5:27_1_2"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="txt" href="#E1:27_1_2_1_2"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="nat_6.html#T14" onmouseover="rs('nat_6/T14')" onmouseout="rh()">NAT_6:14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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