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<b>let </b><font color="Maroon" title="c2">p</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 implies (  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">p</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a>  iff <font color="Maroon" title="c2">p</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> ) )</span><br/>

<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">p</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a>  iff <font color="Maroon" title="c2">p</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">p</font> = <font color="Maroon" title="c2">p</font> as  non <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a> <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">P</font> =  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c3">p</font> as   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> <b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="int_3.html#T21" target="_self" title="INT_3:th.21">Th21</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c3">p</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<div><i><font color="Green" title="E39">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c5">p</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c5">p</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> </span><br/><a NAME="E2:56_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">P</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">P</font> holds <br/> ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">P</font> st <font color="Olive" title="b2">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c4">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">P</font> implies  ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">P</font> st <font color="Olive" title="b1">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c4">P</font> )</span><br/>

<b>assume </b><a NAME="E1:56_1_1"/><i><font color="Green" title="E41">A4</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">P</font> st <font color="Olive" title="b1">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c4">P</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c6">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c5">p</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">a'</font> = <font color="Maroon" title="c7">a'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E4:56_1_1"/>
( 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a'</font> &amp; 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> )
 
;<br/>
<a NAME="E5:56_1_1"/><b>then </b><i><font color="Green" title="E42">A5</font></i>: 
( 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">a'</font> &amp; 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">p</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E6:56_1_1"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">a'</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">p</font> holds <br/><font color="Olive" title="b1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c9">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">p</font> implies <font color="Maroon" title="c9">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 1 )</span><br/>

<b>assume </b><a NAME="E1:56_1_1_1"/><i><font color="Green" title="E43">A6</font></i>: 
( <font color="Maroon" title="c9">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c9">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 1</span><br/>

<b>then consider </b><font color="Maroon" title="c10">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:56_1_1_1"/><i><font color="Green" title="E44">A7</font></i>: 
<font color="Maroon" title="c8">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E4:56_1_1_1"/>
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1_1_1_1"/><i><font color="Green" title="E45">A8</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">a'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_1_1_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:56_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:56_1_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="#E3:56_1_1_1"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:56_1_1_1_1_1_1_2"/><i><font color="Green" title="E46">A9</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:56_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:56_1_1_1_1_1_1_2"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E3:56_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c8">a'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">a'</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E4:56_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:56_1_1_1"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="#E1:56_1_1_1_1"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="#E1:56_1_1_1_1_1_1_2"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:56_1_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:56_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/>
<b>hence </b><a NAME="E6:56_1_1_1"/>
<font color="Maroon" title="c9">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 1
 <b>by </b><i><a class="txt" href="#E1:56_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="#E1:56_1_1_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:56_1_1"/><b>then </b>
<font color="Maroon" title="c8">a'</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="#E5:56_1_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">s</font>, <font color="Maroon" title="c10">t</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E9:56_1_1"/><i><font color="Green" title="E43">A11</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">a'</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_3.html#T16" target="_self" title="INT_3:th.16">Th16</a></i>;<br/>
<a NAME="E10:56_1_1"/><i><font color="Green" title="E44">A12</font></i>: 
( <font color="Maroon" title="c9">s</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c9">s</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font> )
 
<b>by </b><i><a class="ref" href="int_3.html#T9" target="_self" title="INT_3:th.9">Th9</a></i>;<br/>
<a NAME="E11:56_1_1"/><b>then </b>
<font color="Maroon" title="c9">s</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">b'</font> = <font color="Maroon" title="c9">s</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">p</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c5">p</font> <b>by </b><i><a class="txt" href="#E10:56_1_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">b</font> = <font color="Maroon" title="c11">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">P</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">a2</font> = <font color="Maroon" title="c8">a'</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">e</font> = 1 as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<font color="Maroon" title="c12">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a'</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c11">b'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">p</font>
<b>by </b><i><a class="ref" href="int_3.html#D11" target="_self" title="INT_3:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">a2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">p</font>
<b>by </b><i><a class="ref" href="int_3.html#T15" target="_self" title="INT_3:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">a2</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">p</font>
<b>by </b><i><a class="ref" href="int_3.html#T13" target="_self" title="INT_3:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">s</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">p</font>
<b>by </b><i><a class="ref" href="int_3.html#T15" target="_self" title="INT_3:th.15">Th15</a></i>
<br/>.= 
<font color="Maroon" title="c14">e</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">p</font>
<b>by </b><i><a class="txt" href="#E9:56_1_1"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="int_3.html#T8" target="_self" title="INT_3:th.8">Th8</a></i>
<br/>.= 
<font color="Maroon" title="c14">e</font>
<b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="int_3.html#T10" target="_self" title="INT_3:th.10">Th10</a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c4">P</font>
<b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="int_3.html#E46"><i><font color="Green" title="E35">Lm12</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E17:56_1_1"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">P</font> st <font color="Olive" title="b1">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c4">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:56_1"/>
 <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c5">p</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> 
 <b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="int_3.html#T21" target="_self" title="INT_3:th.21">Th21</a>, <a class="ref" href="vectsp_1.html#D20" title="VECTSP_1:def.20">VECTSP_1:def 20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_2"/><i><font color="Green" title="E40">A13</font></i>: 
 <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c5">p</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">p</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a></span><br/><a NAME="E2:56_2"/>
for <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   holds <br/>(  not <font color="Olive" title="b1">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">p</font> or <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">p</font> or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> )</span><br/>

<b>assume </b><a NAME="E1:56_2_1"/>
<font color="Maroon" title="c6">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c7">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:56_2_1"/><i><font color="Green" title="E41">A14</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E4:56_2_1"/><i><font color="Green" title="E42">A15</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_2_1_1"/><i><font color="Green" title="E43">A16</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_2_1_1_1_1"/>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c6">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:56_2_1_1_1_1_1"/>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:56_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:56_2_1"><i><font color="Green" title="E41">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:56_2_1_1_1_1_2"/><i><font color="Green" title="E44">A17</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:56_2_1_1_1_1_2"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:56_2_1_1_1_1_2"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E3:56_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E4:56_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:56_2_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="#E1:56_2_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="#E1:56_2_1_1_1_1_2"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:56_2_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:56_2_1"/><i><font color="Green" title="E43">A19</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_2_1_2"/><i><font color="Green" title="E44">A20</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_2_1_2_1_1"/>
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c7">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:56_2_1_2_1_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:56_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:56_2_1"><i><font color="Green" title="E41">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:56_2_1_2_1_1_2"/><i><font color="Green" title="E45">A21</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:56_2_1_2_1_1_2"/>
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:56_2_1_2_1_1_2"><i><font color="Green" title="E45">A21</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E3:56_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E4:56_2_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:56_2_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="#E1:56_2_1_2"><i><font color="Green" title="E44">A20</font></i></a>, <a class="txt" href="#E1:56_2_1_2_1_1_2"><i><font color="Green" title="E45">A21</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:56_2_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_2_1_3_1"/>
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> or <font color="Maroon" title="c7">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_3_1_1"><b>case </b></a><a NAME="E1:56_2_1_3_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> )</span><br/><div class="add"><a NAME="E2:56_2_1_3_1_1"/><b>then </b>
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="#E3:56_2_1"><i><font color="Green" title="E41">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E3:56_2_1_3_1_1"/>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> )
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_3_1_2"><b>case </b></a><a NAME="E1:56_2_1_3_1_2"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> )</span><br/><div class="add"><a NAME="E2:56_2_1_3_1_2"/><b>then </b><i><font color="Green" title="E44">A23</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="#E4:56_2_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_2_1_3_1_2_1_1"/>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> or <font color="Maroon" title="c6">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_3_1_2_1_1_1"><b>case </b></a><a NAME="E1:56_2_1_3_1_2_1_1_1"/>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> )</span><br/><div class="add"><a NAME="E2:56_2_1_3_1_2_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="#E3:56_2_1"><i><font color="Green" title="E41">A14</font></i></a></i>;<br/><a NAME="E3:56_2_1_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><b>hence </b><a NAME="E4:56_2_1_3_1_2_1_1_1"/>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> )
 <b>by </b><i><a class="txt" href="#E3:56_2_1"><i><font color="Green" title="E41">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_3_1_2_1_1_2"><b>case </b></a><a NAME="E1:56_2_1_3_1_2_1_1_2"/>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:56_2_1_3_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="#E5:56_2_1"><i><font color="Green" title="E43">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">n2</font> = <font color="Maroon" title="c6">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c5">p</font> <b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">n'</font> = <font color="Maroon" title="c8">n2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c5">p</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c10">k2</font> = <font color="Maroon" title="c7">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c5">p</font> <b>by </b><i><a class="txt" href="#E2:56_2_1_3_1_2"><i><font color="Green" title="E44">A23</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">k'</font> = <font color="Maroon" title="c10">k2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c5">p</font></span>)</span> ;<br/><a NAME="E7:56_2_1_3_1_2_1_1_2"/>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c7">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="#E3:56_2_1"><i><font color="Green" title="E41">A14</font></i></a></i>;<br/><a NAME="E8:56_2_1_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E45">A24</font></i>: 
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c5">p</font></span>)</span> &amp; <font color="Maroon" title="c7">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c5">p</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a>, <a class="ref" href="gr_cy_1.html#T12" title="GR_CY_1:th.12">GR_CY_1:12</a></i>;<br/><font color="Maroon" title="c9">n'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">k'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">n2</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">k2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">p</font>
<b>by </b><i><a class="ref" href="int_3.html#D11" target="_self" title="INT_3:def.11">Def11</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E3:56_2_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="int_1.html#T89" title="INT_1:th.89">INT_1:89</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c5">p</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a>, <a class="ref" href="gr_cy_1.html#T12" title="GR_CY_1:th.12">GR_CY_1:12</a></i>
;<br/><b>hence </b><a NAME="E10:56_2_1_3_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:56_2"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="#E8:56_2_1_3_1_2_1_1_2"><i><font color="Green" title="E45">A24</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:56_2_1_3_1_2"/>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:56_2_1"/>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:56_2"/>
<font color="Maroon" title="c5">p</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>
 <b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:56"/>
(  <a href="int_3.html#K9" title="INT_3:func.9">INT.Ring</a> <font color="Maroon" title="c2">p</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a>  iff <font color="Maroon" title="c2">p</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> )
 <b>by </b><i><a class="txt" href="#E5:56"><i><font color="Green" title="E39">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
