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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</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">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">X</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b3">p</font>,<font color="Maroon" title="c2">m</font>) where <font color="Olive" title="b3">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">p</font>,<font color="Maroon" title="c2">m</font>] ) </span> } </span>  &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive" title="b1">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span><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> for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <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> <br/>  for <font color="Olive" title="b3">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b4">X</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b5">p</font>,<font color="Olive" title="b2">m</font>) where <font color="Olive" title="b5">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b5">p</font>,<font color="Olive" title="b2">m</font>] ) </span> } </span>  &amp; <font color="Olive" title="b3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive" title="b3">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b4">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive" title="b3">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>;<br/>
<a NAME="E1:70_1"/><span class="lab"><font color="Green" title="E63">A1</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">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_1_1"/><span class="lab"><font color="Green" title="E64">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</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">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">m</font> being    <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> <br/>  for <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b3">X</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b4">p</font>,<font color="Olive" title="b1">m</font>) where <font color="Olive" title="b4">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b4">p</font>,<font color="Olive" title="b1">m</font>] ) </span> } </span>  &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive" title="b2">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b3">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive" title="b2">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">m</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">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">X</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b3">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b3">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive" title="b1">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive">b<sub>4</sub></font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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">  for <font color="Olive" title="b1">X</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  &amp; <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive">b<sub>3</sub></font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">X</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  &amp; <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> )</span><br/><span class="kw">set </span><font color="Maroon" title="c7">r1</font> = <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/><span class="kw">set </span><font color="Maroon" title="c8">X1</font> = <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span> ;<br/><span class="kw">set </span><font color="Maroon" title="c9">X2</font> = <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span> ;<br/><a NAME="E1:70_1_1_1"/><span class="lab"><font color="Green" title="E65">A3</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">assume </span><a NAME="E2:70_1_1_1"/><span class="lab"><font color="Green" title="E66">A4</font></span>: 
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </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"> ( <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> )</span><br/><span class="kw">assume </span><a NAME="E1:70_1_1_1_1"/><span class="lab"><font color="Green" title="E67">A5</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </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"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1_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="E1:70_1_1_1_1_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_1_1"><span class="lab"><font color="Green" title="E67">A5</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_1_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </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"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">p</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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"> such that </span><br/><a NAME="E3:70_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E68">A6</font></span>: 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c4">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">and </span><br/><a NAME="E4:70_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E69">A7</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E5:70_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E70">A8</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c4">m</font>]
 ;<br/><a NAME="E6:70_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E7:70_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:70_1_1_1_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1_1"><span class="lab"><font color="Green" title="E66">A4</font></span></a>, <a class="txt" href="#E3:70_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="txt" href="#E5:70_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E70">A8</font></span></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="70_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_1_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </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"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font></span><br/><div class="add"><a NAME="E2:70_1_1_1_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  st <br/>( <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] )
 ;<br/><span class="kw">hence </span><a NAME="E3:70_1_1_1_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1_1"><span class="lab"><font color="Green" title="E66">A4</font></span></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><span class="kw">end;</span></div><a NAME="E4:70_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A9</font></span>: 
<span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font>
 ;<br/><span class="kw">assume </span><a NAME="E5:70_1_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_2"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  )</span><br/><span class="kw">assume </span><a NAME="E1:70_1_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">p</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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"> such that </span><br/><a NAME="E3:70_1_1_1_2"/><span class="lab"><font color="Green" title="E68">A10</font></span>: 
( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c4">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c4">m</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1_1"><span class="lab"><font color="Green" title="E66">A4</font></span></a></span>;<br/><a NAME="E4:70_1_1_1_2"/>
( ( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c4">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c4">m</font>] ) or ( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c4">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c4">m</font>] ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1_1_2"><span class="lab"><font color="Green" title="E68">A10</font></span></a></span>;<br/><a NAME="E5:70_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  )
 ;<br/><span class="kw">hence </span><a NAME="E6:70_1_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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><a NAME="E7:70_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span> 
 ;<br/><a NAME="E8:70_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A11</font></span>: 
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">X1</font> = <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finset_1.html#T1" onmouseover="rs('finset_1/T1')" onmouseout="rh()">FINSET_1:1</a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">assume </span><a NAME="E10:70_1_1_1"/><span class="lab"><font color="Green" title="E69">A12</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</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">  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span><br/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A13</font></span>: <font color="Maroon" title="c3">n</font> = 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>

<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T28" onmouseover="rs('int_1/T28')" onmouseout="rh()">INT_1:28</a></span>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>

;<br/><a NAME="E12:70_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A14</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D6" onmouseover="rs('int_1/D6')" onmouseout="rh()">INT_1:def 6</a></span>;<br/><a NAME="E13:70_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A15</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">X1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E11:70_1_1_1"><span class="lab"><font color="Green" title="E70">A13</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_4"><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="E1:70_1_1_1_4"/>
(  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  or  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_4_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_1_4_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </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">  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span><br/><div class="add"><a NAME="E2:70_1_1_1_4_1"/><span class="kw">then </span>
<span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:70_1_1_1_4_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_1_1_1"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="txt" href="#E11:70_1_1_1"><span class="lab"><font color="Green" title="E70">A13</font></span></a>, <a class="txt" href="#E8:70_1_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E13:70_1_1_1"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="txt" href="#E1:70_1_1_1"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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="70_1_1_1_4_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_1_4_2"/><span class="lab"><font color="Green" title="E73">A16</font></span>: 
 ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </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">  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:70_1_1_1_4_2"/><span class="lab"><font color="Green" title="E74">A17</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span> 
 ;<br/><div><span class="lab"><font color="Green" title="E75">A18</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_4_2_1"><span class="kw">now </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">  not <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:70_1_1_1_4_2_1"/>
<span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:70_1_1_1_4_2_1"/><span class="lab"><font color="Green" title="E76">A19</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E4:70_1_1_1_4_2_1"/><span class="lab"><font color="Green" title="E77">A20</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_1_4_2"><span class="lab"><font color="Green" title="E73">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T35" onmouseover="rs('zfmisc_1/T35')" onmouseout="rh()">ZFMISC_1:35</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">p1</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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"> such that </span><br/><a NAME="E6:70_1_1_1_4_2_1"/><span class="lab"><font color="Green" title="E78">A21</font></span>: 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c13">p1</font>,<font color="Maroon" title="c4">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">and </span><br/><a NAME="E7:70_1_1_1_4_2_1"/><span class="lab"><font color="Green" title="E79">A22</font></span>: 
( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">p1</font> &amp; <font color="Maroon" title="c13">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> )
 <span class="kw">and </span><br/><a NAME="E8:70_1_1_1_4_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c13">p1</font>,<font color="Maroon" title="c4">m</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1_1_4_2"><span class="lab"><font color="Green" title="E74">A17</font></span></a></span>;<br/><a NAME="E9:70_1_1_1_4_2_1"/><span class="lab"><font color="Green" title="E80">A23</font></span>: 
<font color="Maroon" title="c13">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1_1_1_4_2_1"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="int_1.html#D6" onmouseover="rs('int_1/D6')" onmouseout="rh()">INT_1:def 6</a></span>;<br/><a NAME="E10:70_1_1_1_4_2_1"/>
 ex <font color="Olive" title="b1">p2</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  st <br/>( <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p2</font>,<font color="Maroon" title="c4">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font> &amp; <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p2</font> &amp; <font color="Olive" title="b1">p2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p2</font>,<font color="Maroon" title="c4">m</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1_1_4_2_1"><span class="lab"><font color="Green" title="E76">A19</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:70_1_1_1_4_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_1_4_2_1"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E6:70_1_1_1_4_2_1"><span class="lab"><font color="Green" title="E78">A21</font></span></a>, <a class="txt" href="#E9:70_1_1_1_4_2_1"><span class="lab"><font color="Green" title="E80">A23</font></span></a>, <a class="ref" href="int_1.html#D6" onmouseover="rs('int_1/D6')" onmouseout="rh()">INT_1:def 6</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="70_1_1_1_4_2_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="E1:70_1_1_1_4_2_2"/>
(  not <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">X1</font> or <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">X1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_4_2_2_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_1_4_2_2_1"/>
 not <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">X1</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">  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span><br/><div class="add"><a NAME="E2:70_1_1_1_4_2_2_1"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c10">X1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_1_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E4:70_1_1_1_4_2"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="ref" href="card_2.html#T41" onmouseover="rs('card_2/T41')" onmouseout="rh()">CARD_2:41</a></span>;<br/><span class="kw">hence </span><a NAME="E3:70_1_1_1_4_2_2_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_1_1_1"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="txt" href="#E11:70_1_1_1"><span class="lab"><font color="Green" title="E70">A13</font></span></a>, <a class="txt" href="#E13:70_1_1_1"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</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="70_1_1_1_4_2_2_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_1_4_2_2_2"/>
<font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">X1</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">  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span><br/><div class="add"><a NAME="E2:70_1_1_1_4_2_2_2"/><span class="kw">then </span>
<span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_1_4_2"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><a NAME="E3:70_1_1_1_4_2_2_2"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E11:70_1_1_1"><span class="lab"><font color="Green" title="E70">A13</font></span></a>, <a class="txt" href="#E12:70_1_1_1"><span class="lab"><font color="Green" title="E71">A14</font></span></a>, <a class="txt" href="#E8:70_1_1_1"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:70_1_1_1_4_2_2_2"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_1_1_1"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="txt" href="#E1:70_1_1_1"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:70_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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>
<a NAME="E2:70_1"/><span class="lab"><font color="Green" title="E64">A24</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</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">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">X</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b3">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b3">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive" title="b1">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Olive" title="b1">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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">  for <font color="Olive" title="b1">X</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  &amp; <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">X</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </span>  &amp; <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:70_1_2"/><span class="lab"><font color="Green" title="E65">A25</font></span>: 
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">m</font>] ) </span> } </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"> (  not <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  not  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> or  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:70_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> or  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:70_1_2"/><span class="lab"><font color="Green" title="E66">A26</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</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">  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span><br/>

<a NAME="E4:70_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D6" onmouseover="rs('int_1/D6')" onmouseout="rh()">INT_1:def 6</a></span>;<br/>
<a NAME="E5:70_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A27</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_2_1"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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 <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font></span><br/><span class="kw">assume </span><a NAME="E1:70_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">p</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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"> such that </span><br/><a NAME="E3:70_1_2_1"/>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c4">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">and </span><br/><a NAME="E4:70_1_2_1"/><span class="lab"><font color="Green" title="E68">A28</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">and </span><br/><a NAME="E5:70_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c4">m</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_2"><span class="lab"><font color="Green" title="E65">A25</font></span></a></span>;<br/><a NAME="E6:70_1_2_1"/>
<font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_2"><span class="lab"><font color="Green" title="E67">A27</font></span></a>, <a class="txt" href="#E4:70_1_2_1"><span class="lab"><font color="Green" title="E68">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:70_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</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>
<span class="kw">hence </span><a NAME="E7:70_1_2"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_2"><span class="lab"><font color="Green" title="E66">A26</font></span></a>, <a class="ref" href="card_1.html#T27" onmouseover="rs('card_1/T27')" onmouseout="rh()">CARD_1:27</a>, <a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</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>
<a NAME="E3:70_1"/><span class="lab"><font color="Green" title="E65">A29</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E64">A24</font></span></a>, <a class="txt" href="#E1:70_1"><span class="lab"><font color="Green" title="E63">A1</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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">  for <font color="Olive" title="b1">X</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c2">m</font>) where <font color="Olive" title="b2">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c2">m</font>] ) </span> } </span>  &amp; <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">m</font>] ) </span> } </span>  &amp; <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:70_1"/><span class="lab"><font color="Green" title="E66">A30</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">m</font>) where <font color="Olive" title="b1">p</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  : ( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">m</font>] ) </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E5:70_1"/><span class="lab"><font color="Green" title="E67">A31</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span><br/>

<a NAME="E6:70_1"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> 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="txt" href="#E5:70_1"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="ref" href="int_1.html#T53" onmouseover="rs('int_1/T53')" onmouseout="rh()">INT_1:53</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:70_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1"><span class="lab"><font color="Green" title="E65">A29</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="txt" href="#E5:70_1"><span class="lab"><font color="Green" title="E67">A31</font></span></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>
