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<span class="kw">given </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> being   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a><span class="kw"> such that </span><a NAME="E2:125"/><span class="lab"><font color="Green" title="E84">A1</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7
 ; <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/>

<span class="kw">consider </span><font color="Maroon" title="c4">n1</font>, <font color="Maroon" title="c5">m1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E4:125"/><span class="lab"><font color="Green" title="E85">A2</font></span>: 
<font color="Maroon" title="c5">m1</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">and </span><br/><a NAME="E5:125"/><span class="lab"><font color="Green" title="E86">A3</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rat_1.html#T2" onmouseover="rs('rat_1/T2')" onmouseout="rh()">RAT_1:2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">n2</font>, <font color="Maroon" title="c7">m2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E7:125"/><span class="lab"><font color="Green" title="E87">A4</font></span>: 
<font color="Maroon" title="c7">m2</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">and </span><br/><a NAME="E8:125"/><span class="lab"><font color="Green" title="E88">A5</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n2</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rat_1.html#T2" onmouseover="rs('rat_1/T2')" onmouseout="rh()">RAT_1:2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">n3</font>, <font color="Maroon" title="c9">m3</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E10:125"/><span class="lab"><font color="Green" title="E89">A6</font></span>: 
<font color="Maroon" title="c9">m3</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">and </span><br/><a NAME="E11:125"/><span class="lab"><font color="Green" title="E90">A7</font></span>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n3</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">m3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rat_1.html#T2" onmouseover="rs('rat_1/T2')" onmouseout="rh()">RAT_1:2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">m</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">m2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m3</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">m</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">m2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m3</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="#E4:125"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E7:125"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E10:125"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">a</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">m2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m3</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">n2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">m1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m3</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">c</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c8">n3</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">m1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">m2</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st 7 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">$1 <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> );<br/>
<a NAME="E13:125"/><span class="lab"><font color="Green" title="E91">A8</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">m1</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">n2</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">m2</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n2</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m2</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">n3</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">m3</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n3</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m3</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T76" onmouseover="rs('xcmplx_1/T76')" onmouseout="rh()">XCMPLX_1:76</a></span>;<br/>
<a NAME="E14:125"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n2</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">m2</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">m2</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n2</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">m2</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:125"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E7:125"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="ref" href="xcmplx_1.html#T116" onmouseover="rs('xcmplx_1/T116')" onmouseout="rh()">XCMPLX_1:116</a></span>;<br/>
<span class="kw">then </span>7 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">m2</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">n2</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m3</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">n3</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">m2</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">m2</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m3</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E4:125"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E5:125"><span class="lab"><font color="Green" title="E86">A3</font></span></a>, <a class="txt" href="#E7:125"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E8:125"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="txt" href="#E10:125"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E11:125"><span class="lab"><font color="Green" title="E90">A7</font></span></a>, <a class="txt" href="#E13:125"><span class="lab"><font color="Green" title="E91">A8</font></span></a>, <a class="ref" href="xcmplx_1.html#T116" onmouseover="rs('xcmplx_1/T116')" onmouseout="rh()">XCMPLX_1:116</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">m2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m3</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">n2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">m1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m3</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">n3</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">m1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">m2</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>

;<br/>
<a NAME="E16:125"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">m2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m3</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">n2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">m1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">m3</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">n3</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">m1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">m2</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:125"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E7:125"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E10:125"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<a NAME="E17:125"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A9</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:125"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E7:125"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E10:125"><span class="lab"><font color="Green" title="E89">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">M</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E19:125"/><span class="lab"><font color="Green" title="E93">A10</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">M</font>]
 <span class="kw">and </span><br/><a NAME="E20:125"/><span class="lab"><font color="Green" title="E94">A11</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c15">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" onmouseover="rs('nat_1/S5')" onmouseout="rh()">NAT_1:sch 5</a>(<a class="txt" href="#E17:125"><span class="lab"><font color="Green" title="E92">A9</font></span></a>);<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c16">a</font>, <font color="Maroon" title="c17">b</font>, <font color="Maroon" title="c18">c</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E22:125"/><span class="lab"><font color="Green" title="E95">A12</font></span>: 
7 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">M</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">c</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:125"><span class="lab"><font color="Green" title="E93">A10</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2"><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="E1:125_2"/>
( <font color="Maroon" title="c15">M</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c15">M</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_2_1"><span class="kw">suppose </span></a><a NAME="E1:125_2_1"/>
<font color="Maroon" title="c15">M</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:125_2_1"/><span class="lab"><font color="Green" title="E96">A13</font></span>: 
<font color="Maroon" title="c15">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">n</font>
 ;<br/><a NAME="E4:125_2_1"/><span class="lab"><font color="Green" title="E97">A14</font></span>: 
<span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">7 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 4 <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="nat_d.html#T13" onmouseover="rs('nat_d/T13')" onmouseout="rh()">NAT_D:13</a></span>;<br/><a NAME="E5:125_2_1"/>
<font color="Maroon" title="c16">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:125_2_1"><span class="lab"><font color="Green" title="E96">A13</font></span></a>, <a class="txt" href="#E22:125"><span class="lab"><font color="Green" title="E95">A12</font></span></a>, <a class="txt" href="#E4:125_2_1"><span class="lab"><font color="Green" title="E97">A14</font></span></a>, <a class="ref" href="number14.html#T91" target="_self" onmouseover="rs('number14/T91')" onmouseout="rh()">Th91</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">a1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E7:125_2_1"/><span class="lab"><font color="Green" title="E98">A15</font></span>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T11" onmouseover="rs('abian/T11')" onmouseout="rh()">ABIAN:11</a></span>;<br/><a NAME="E8:125_2_1"/>
<font color="Maroon" title="c17">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:125_2_1"><span class="lab"><font color="Green" title="E96">A13</font></span></a>, <a class="txt" href="#E22:125"><span class="lab"><font color="Green" title="E95">A12</font></span></a>, <a class="txt" href="#E4:125_2_1"><span class="lab"><font color="Green" title="E97">A14</font></span></a>, <a class="ref" href="number14.html#T91" target="_self" onmouseover="rs('number14/T91')" onmouseout="rh()">Th91</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">b1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E10:125_2_1"/><span class="lab"><font color="Green" title="E99">A16</font></span>: 
<font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c21">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T11" onmouseover="rs('abian/T11')" onmouseout="rh()">ABIAN:11</a></span>;<br/><a NAME="E11:125_2_1"/>
<font color="Maroon" title="c18">c</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:125_2_1"><span class="lab"><font color="Green" title="E96">A13</font></span></a>, <a class="txt" href="#E22:125"><span class="lab"><font color="Green" title="E95">A12</font></span></a>, <a class="txt" href="#E4:125_2_1"><span class="lab"><font color="Green" title="E97">A14</font></span></a>, <a class="ref" href="number14.html#T91" target="_self" onmouseover="rs('number14/T91')" onmouseout="rh()">Th91</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">c1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E13:125_2_1"/><span class="lab"><font color="Green" title="E100">A17</font></span>: 
<font color="Maroon" title="c18">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c22">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T11" onmouseover="rs('abian/T11')" onmouseout="rh()">ABIAN:11</a></span>;<br/><a NAME="E14:125_2_1"/><span class="lab"><font color="Green" title="E101">A18</font></span>: 
<font color="Maroon" title="c19">n</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E19:125"><span class="lab"><font color="Green" title="E93">A10</font></span></a>, <a class="txt" href="#E3:125_2_1"><span class="lab"><font color="Green" title="E96">A13</font></span></a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">a1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">b1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">c1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:125_2_1"><span class="lab"><font color="Green" title="E98">A15</font></span></a>, <a class="txt" href="#E10:125_2_1"><span class="lab"><font color="Green" title="E99">A16</font></span></a>, <a class="txt" href="#E13:125_2_1"><span class="lab"><font color="Green" title="E100">A17</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default">7 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">M</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:125"><span class="lab"><font color="Green" title="E95">A12</font></span></a></span>
<br/>.= 
7 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:125_2_1"><span class="lab"><font color="Green" title="E96">A13</font></span></a></span>
;<br/><a NAME="E16:125_2_1"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:125_2_1"><span class="lab"><font color="Green" title="E101">A18</font></span></a>, <a class="txt" href="#E20:125"><span class="lab"><font color="Green" title="E94">A11</font></span></a>, <a class="txt" href="#E3:125_2_1"><span class="lab"><font color="Green" title="E96">A13</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E17:125_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:125_2_1"><span class="lab"><font color="Green" title="E101">A18</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</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="125_2_2"><span class="kw">suppose </span></a><a NAME="E1:125_2_2"/>
<font color="Maroon" title="c15">M</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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"> contradiction</span><br/><div class="add"><a NAME="E2:125_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">M</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number05.html#T41" onmouseover="rs('number05/T41')" onmouseout="rh()">NUMBER05:41</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default">7 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">M</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 8
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T67" onmouseover="rs('nat_d/T67')" onmouseout="rh()">NAT_D:67</a></span>
<br/>.= 
7
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:125_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:125"><span class="lab"><font color="Green" title="E95">A12</font></span></a>, <a class="ref" href="number14.html#T92" target="_self" onmouseover="rs('number14/T92')" onmouseout="rh()">Th92</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>
