let O be set ; :: thesis: for G being strict Group ex H being non empty HGrWOpStr of O st
( H is strict & H is distributive & H is Group-like & H is associative & G = multMagma(# the carrier of H,the multF of H #) )
let G be strict Group; :: thesis: ex H being non empty HGrWOpStr of O st
( H is strict & H is distributive & H is Group-like & H is associative & G = multMagma(# the carrier of H,the multF of H #) )
reconsider h = [:O,{(id the carrier of G)}:] as Action of O,the carrier of G by Lm2;
set A = the carrier of G;
set m = the multF of G;
set GO = HGrWOpStr(# the carrier of G,the multF of G,h #);
reconsider GO = HGrWOpStr(# the carrier of G,the multF of G,h #) as non empty HGrWOpStr of O ;
reconsider G9 = GO as non empty multMagma ;

A1: now

set e = 1_ G;
reconsider e9 = 1_ G as Element of G9 ;
take e9 = e9; :: thesis: for h9 being Element of G9 holds
( h9 * e9 = h9 & e9 * h9 = h9 & ex g9 being Element of G9 st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
let h9 be Element of G9; :: thesis: ( h9 * e9 = h9 & e9 * h9 = h9 & ex g9 being Element of G9 st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
reconsider h = h9 as Element of G ;
set g = h " ;
reconsider g9 = h " as Element of G9 ;
h9 * e9 = h * (1_ G)
.= h by GROUP_1:def 5 ;
hence h9 * e9 = h9 ; :: thesis: ( e9 * h9 = h9 & ex g9 being Element of G9 st
( h9 * g9 = e9 & g9 * h9 = e9 ) )
e9 * h9 = (1_ G) * h
.= h by GROUP_1:def 5 ;
hence e9 * h9 = h9 ; :: thesis: ex g9 being Element of G9 st
( h9 * g9 = e9 & g9 * h9 = e9 )
take g9 = g9; :: thesis: ( h9 * g9 = e9 & g9 * h9 = e9 )
h9 * g9 = h * (h " )
.= 1_ G by GROUP_1:def 6 ;
hence h9 * g9 = e9 ; :: thesis: g9 * h9 = e9
g9 * h9 = (h " ) * h
.= 1_ G by GROUP_1:def 6 ;
hence g9 * h9 = e9 ; :: thesis: verum

end;

take GO ; :: thesis: ( GO is strict & GO is distributive & GO is Group-like & GO is associative & G = multMagma(# the carrier of GO,the multF of GO #) )

A2: now

let G99 be Group; :: thesis: for a being Action of O,the carrier of G99 st a = the action of GO & multMagma(# the carrier of G99,the multF of G99 #) = multMagma(# the carrier of GO,the multF of GO #) holds
a is distributive
let a be Action of O,the carrier of G99; :: thesis: ( a = the action of GO & multMagma(# the carrier of G99,the multF of G99 #) = multMagma(# the carrier of GO,the multF of GO #) implies a is distributive )
assume A3: a = the action of GO ; :: thesis: ( multMagma(# the carrier of G99,the multF of G99 #) = multMagma(# the carrier of GO,the multF of GO #) implies a is distributive )
assume A4: multMagma(# the carrier of G99,the multF of G99 #) = multMagma(# the carrier of GO,the multF of GO #) ; :: thesis: a is distributive

now

let o be Element of O; :: thesis: ( o in O implies a . o is Homomorphism of G99,G99 )
assume o in O ; :: thesis: a . o is Homomorphism of G99,G99
then o in dom h by FUNCT_2:def 1;
then [o,(h . o)] in [:O,{(id the carrier of G99)}:] by A4, FUNCT_1:8;
then consider x, y being set such that
x in O and
A5: ( y in {(id the carrier of G99)} & [o,(h . o)] = [x,y] ) by ZFMISC_1:def 2;
( y = id the carrier of G99 & h . o = y ) by A5, TARSKI:def 1, ZFMISC_1:33;
hence a . o is Homomorphism of G99,G99 by A3, GROUP_6:47; :: thesis: verum

end;

hence a is distributive by Def4; :: thesis: verum

end;

now

let x9, y9, z9 be Element of G9; :: thesis: (x9 * y9) * z9 = x9 * (y9 * z9)
reconsider x = x9, y = y9, z = z9 as Element of G ;
(x9 * y9) * z9 = (x * y) * z
.= x * (y * z) by GROUP_1:def 4 ;
hence (x9 * y9) * z9 = x9 * (y9 * z9) ; :: thesis: verum

end;

hence ( GO is strict & GO is distributive & GO is Group-like & GO is associative & G = multMagma(# the carrier of GO,the multF of GO #) ) by A1, A2, Def5, GROUP_1:def 3, GROUP_1:def 4; :: thesis: verum