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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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">  <a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pcomps_1.html#D6" onmouseover="rs('pcomps_1/D6')" onmouseout="rh()">PCOMPS_1:def 6</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"> ( (  not <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> ) &amp; (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> or <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp; <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font>) &amp; <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>)</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"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>)</span>)</span> )</span><br/>



<a NAME="E1:38"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">z</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D6" target="_self" onmouseover="rs('euclid/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E2:38"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D6" target="_self" onmouseover="rs('euclid/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:38"/>
( <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  iff <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T16" target="_self" onmouseover="rs('euclid/T16')" onmouseout="rh()">Th13</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"> ( <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font>) &amp; <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>)</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"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>)</span>)</span> )</span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D6" target="_self" onmouseover="rs('euclid/D6')" onmouseout="rh()">Def6</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T18" target="_self" onmouseover="rs('euclid/T18')" onmouseout="rh()">Th15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D6" target="_self" onmouseover="rs('euclid/D6')" onmouseout="rh()">Def6</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"> <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>)</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"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>)</span>)</span></span><br/>

<a NAME="E5:38"/>
( <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">z</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D6" target="_self" onmouseover="rs('euclid/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:38"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>)</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="euclid.html#T19" target="_self" onmouseover="rs('euclid/T19')" onmouseout="rh()">Th16</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/>


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