<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">Gc</font> be   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span>,<font color="Maroon" title="c1">Gc</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">n</font> =  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>;<br/>
<a NAME="E1:22"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font><span class="kw"> such that </span><br/><a NAME="E3:22"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font>  ex <font color="Olive" title="b2">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_1.html#T42" title="GR_CY_1:th.42">GR_CY_1:42</a></span>;<br/>
<a NAME="E4:22"/>
for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font>  ex <font color="Olive" title="b2">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">i</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E2:22_1"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E16">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">p1</font> = <font color="Maroon" title="c5">p</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">p1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">p1</font></span><br/>

<span class="kw">thus </span><a NAME="E4:22_1"/>
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c1">Gc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_2.html#T11" target="_self" title="GR_CY_2:th.11">Th11</a></span>;<br/>
<a NAME="E6:22"/>
 ex <font color="Olive" title="b1">h</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">Gc</font> st <font color="Olive" title="b1">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font> = <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> $1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font><span class="kw"> such that </span><br/><a NAME="E2:22_2"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span> holds  <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">p</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<a NAME="E3:22_2"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
 <a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K3" title="GR_CY_1:func.3">addint</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="gr_cy_1.html#D6" title="GR_CY_1:def.6">GR_CY_1:def 6</a></span>;<br/>
<a NAME="E4:22_2"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">j1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span> holds  <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c6">j1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">j1</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">j9</font> = <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c8">j19</font> = <font color="Maroon" title="c6">j1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2"><span class="lab"><font color="Green" title="E19">A6</font></span></a></span>;<br/>
<a NAME="E2:22_2_1"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self" title="GR_CY_2:def.1">Def1</a></span>;<br/>
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">j1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K3" title="GR_CY_1:func.3">addint</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">j9</font>,<font color="Maroon" title="c8">j19</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self" title="GR_CY_2:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j9</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">j19</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="gr_cy_1.html#D5" title="GR_CY_1:def.5">GR_CY_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:22_2_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self" title="GR_CY_2:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="newton.html#T77" title="NEWTON:th.77">NEWTON:77</a></span>
;<br/>
<span class="kw">then </span><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">j1</font></span>)</span> = 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T70" title="GROUP_1:th.70">GROUP_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p4">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T67" title="GROUP_1:th.67">GROUP_1:67</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_1.html#T29" title="GR_CY_1:th.29">GR_CY_1:29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T61" title="GROUP_1:th.61">GROUP_1:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T16" title="GROUP_1:th.16">GROUP_1:16</a></span>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c6">j1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">j1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:22_2_1"/>
<font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">j1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:22_2"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<font color="Maroon" title="c4">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c4">h</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c4">h</font> or  not <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c4">h</font> or  not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c4">h</font> or  not <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_2_2"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">h</font>
 <span class="kw">and </span><br/><a NAME="E2:22_2_2"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">h</font>
 <span class="kw">and </span><br/><a NAME="E3:22_2_2"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">and </span><br/><a NAME="E4:22_2_2"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">y9</font> = <font color="Maroon" title="c6">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x9</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_2"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E26">A14</font></span>: <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c8">x9</font></span>)</span> = 
<font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E3:22_2_2"><span class="lab"><font color="Green" title="E24">A12</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c7">y9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>
;<br/>
<a NAME="E8:22_2_2"/>
 ex <font color="Olive" title="b1">p</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#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c9">p</font> =  <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c8">x9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">m</font> =  <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c7">y9</font>;<br/>
<a NAME="E1:22_2_2_2"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c7">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self" title="GR_CY_2:def.1">Def1</a></span>;<br/>
<a NAME="E2:22_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A16</font></span>: 
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c7">y9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E3:22_2_2_2"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c8">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self" title="GR_CY_2:def.1">Def1</a></span>;<br/>
<a NAME="E4:22_2_2_2"/><span class="kw">then </span>
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c8">x9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:22_2_2_2"/>
 ex <font color="Olive" title="b1">p</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#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2_2"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E7:22_2_2"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E3:22_2_2_2"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="txt" href="#E1:22_2_2_2"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E2:22_2_2_2"><span class="lab"><font color="Green" title="E28">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E10:22_2_2"/><span class="lab"><font color="Green" title="E27">A18</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">and </span><br/><a NAME="E11:22_2_2"/><span class="lab"><font color="Green" title="E28">A19</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">and </span><br/><a NAME="E12:22_2_2"/><span class="lab"><font color="Green" title="E29">A20</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>
 <span class="kw">and </span><br/><a NAME="E13:22_2_2"/><span class="lab"><font color="Green" title="E30">A21</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>
 ;<br/>
<a NAME="E14:22_2_2"/><span class="lab"><font color="Green" title="E31">A22</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_2_2_3_1_1"/>
( <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> or <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">p</font> or <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:22_2_2_3_1_1_1"/><span class="lab"><font color="Green" title="E32">A23</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">m1</font> = <font color="Maroon" title="c10">m</font>, <font color="Maroon" title="c12">p1</font> = <font color="Maroon" title="c9">p</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">r1</font> = <font color="Maroon" title="c11">m1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">p1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_2_3_1_1_1"><span class="lab"><font color="Green" title="E32">A23</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2_3_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_2_2_3_1_1_1_1_1"/>
( <font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c13">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2_3_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_2_2_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E33">A24</font></span>: 
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c14">zero</font> =  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ;<br/><a NAME="E2:22_2_2_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A25</font></span>: 
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2_3_1_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c15">n1</font> =  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E2:22_2_2_3_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">m1</font> <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> <font color="Maroon" title="c12">p1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22_2_2"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:22_2_2_3_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:22_2_2_3_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c11">m1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">p1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">p1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22_2_2"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></span>;<br/><a NAME="E4:22_2_2_3_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">p1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></span>;<br/><span class="kw">hence </span><a NAME="E5:22_2_2_3_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_2_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E33">A24</font></span></a>, <a class="txt" href="#E2:22_2_2_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A25</font></span></a>, <a class="ref" href="group_1.html#T43" title="GROUP_1:th.43">GROUP_1:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="22_2_2_3_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_2_2_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c13">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:22_2_2_3_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_2_2"><span class="lab"><font color="Green" title="E27">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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><span class="kw">hence </span><a NAME="E5:22_2_2_3_1_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="22_2_2_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:22_2_2_3_1_1_2"/><span class="lab"><font color="Green" title="E32">A26</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">m1</font> = <font color="Maroon" title="c10">m</font>, <font color="Maroon" title="c12">p1</font> = <font color="Maroon" title="c9">p</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">r1</font> = <font color="Maroon" title="c12">p1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">m1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_2_3_1_1_2"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2_3_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_2_2_3_1_1_2_1"/>
( <font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c13">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2_3_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_2_2_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E33">A27</font></span>: 
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c14">zero</font> =  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ;<br/><a NAME="E2:22_2_2_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E34">A28</font></span>: 
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2_3_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c15">n1</font> =  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E2:22_2_2_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c12">p1</font> <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> <font color="Maroon" title="c11">m1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_2_2"><span class="lab"><font color="Green" title="E29">A20</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:22_2_2_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c13">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:22_2_2_3_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c12">p1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c11">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c11">m1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:22_2_2"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></span>;<br/><a NAME="E4:22_2_2_3_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c11">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></span>;<br/><span class="kw">hence </span><a NAME="E5:22_2_2_3_1_1_2_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_2_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="txt" href="#E2:22_2_2_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="ref" href="group_1.html#T43" title="GROUP_1:th.43">GROUP_1:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="22_2_2_3_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_2_2_3_1_1_2_1_2"/>
<font color="Maroon" title="c13">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:22_2_2_3_1_1_2_1_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_2_2"><span class="lab"><font color="Green" title="E27">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2_3_1_1_3"><span class="kw">case </span></a><a NAME="E1:22_2_2_3_1_1_3"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:22_2_2_3_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22_2_2"><span class="lab"><font color="Green" title="E27">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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>
<span class="kw">hence </span><a NAME="E2:22_2_2_3"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> &amp; <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:22_2_2"/>
( not <font color="Maroon" title="c3">g</font> is  <a href="group_1.html#V4" title="GROUP_1:attr.4">being_of_order_0</a>  &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="gr_cy_1.html#T26" title="GR_CY_1:th.26">GR_CY_1:26</a>, <a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:22_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_2_2"><span class="lab"><font color="Green" title="E31">A22</font></span></a>, <a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:22_2"/><span class="lab"><font color="Green" title="E22">A29</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:22_2"/><span class="lab"><font color="Green" title="E23">A30</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_2_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:22_2_3"/><span class="lab"><font color="Green" title="E24">A31</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E16">A2</font></span></a></span>;<br/>
<a NAME="E5:22_2_3"/>
 ex <font color="Olive" title="b1">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">h</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c8">t</font> = <font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span>;<br/>
<a NAME="E1:22_2_3_1"/>
<font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">t9</font> = <font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_2"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">t9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">t9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">h</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">t9</font> )</span><br/>

<a NAME="E3:22_2_3_1"/><span class="lab"><font color="Green" title="E25">A32</font></span>: 
 <a href="gr_cy_2.html#K1" title="GR_CY_2:func.1">@</a> <font color="Maroon" title="c9">t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self" title="GR_CY_2:def.1">Def1</a></span>;<br/>
<font color="Maroon" title="c6">z</font> = 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E4:22_2_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T67" title="GROUP_1:th.67">GROUP_1:67</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_1.html#T29" title="GR_CY_1:th.29">GR_CY_1:29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">Gc</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T61" title="GROUP_1:th.61">GROUP_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:22_2_3_1"/>
( <font color="Maroon" title="c9">t9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">h</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">t9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_2"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E6:22_2"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E3:22_2_3_1"><span class="lab"><font color="Green" title="E25">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:22_2_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:22_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E9:22_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A33</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Gc</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">h</font> = <font color="Maroon" title="c4">h</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">Gc</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">h</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<a NAME="E11:22_2"/>
<font color="Maroon" title="c5">h</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_2"><span class="lab"><font color="Green" title="E24">A33</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:22_2"/>
<font color="Maroon" title="c5">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_2"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:22"/>
 <a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font></span>)</span>,<font color="Maroon" title="c1">Gc</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D15" title="GROUP_6:def.15">GROUP_6:def 15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
