A1: ex e being Element of (Group_of_Perm n) st
for p being Element of (Group_of_Perm n) holds
( p * e = p & e * p = p & ex g being Element of (Group_of_Perm n) st
( p * g = e & g * p = e ) )

reconsider e = idseq n as Element of (Group_of_Perm n) by Th5;
take e ; :: thesis:
reconsider e9 = idseq n as Element of Permutations n by Def5;
A2: for p being Element of (Group_of_Perm n) ex g being Element of (Group_of_Perm n) st
( p * g = e & g * p = e )

let p be Element of (Group_of_Perm n); :: thesis:
reconsider q = p as Element of Permutations n by Def6;
reconsider r = q as Permutation of (Seg n) by Def5;
reconsider f = r " as Element of Permutations n by Def5;
reconsider g = f as Element of (Group_of_Perm n) by Th8;
take g ; :: thesis:
A3: g * p = q * f by Def6
.= e by Th7 ;
p * g = f * q by Def6
.= e by Th7 ;
hence ( p * g = e & g * p = e ) by A3; :: thesis:

end;

for p being Element of (Group_of_Perm n) holds
( p * e = p & e * p = p )

let p be Element of (Group_of_Perm n); :: thesis:
reconsider p9 = p as Element of Permutations n by Def6;
A4: e * p = p9 * e9 by Def6
.= p by Th6 ;
p * e = e9 * p9 by Def6
.= p by Th6 ;
hence ( p * e = p & e * p = p ) by A4; :: thesis:

end;

hence for p being Element of (Group_of_Perm n) holds
( p * e = p & e * p = p & ex g being Element of (Group_of_Perm n) st
( p * g = e & g * p = e ) ) by A2; :: thesis:

end;

let p, q, r be Element of (Group_of_Perm n); :: thesis:
reconsider p9 = p, q9 = q, r9 = r as Element of Permutations n by Def6;
A5: ( q9 * p9 is Permutation of (Seg n) & r9 * q9 is Permutation of (Seg n) )

reconsider p9 = p9, q9 = q9, r9 = r9 as Permutation of (Seg n) by Def5;
( q9 * p9 is Permutation of (Seg n) & r9 * q9 is Permutation of (Seg n) ) ;
hence ( q9 * p9 is Permutation of (Seg n) & r9 * q9 is Permutation of (Seg n) ) ; :: thesis:

end;

then A6: q9 * p9 is Element of Permutations n by Def5;
A7: r9 * q9 is Element of Permutations n by A5, Def5;
(p * q) * r = the multF of (Group_of_Perm n) . ((q9 * p9),r9) by Def6
.= r9 * (q9 * p9) by A6, Def6
.= (r9 * q9) * p9 by RELAT_1:36
.= the multF of (Group_of_Perm n) . (p9,(r9 * q9)) by A7, Def6
.= p * (q * r) by Def6 ;
hence (p * q) * r = p * (q * r) ; :: thesis:

end;