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

proof

reconsider e = idseq n as Element of by Th23;
take e ; :: thesis: for p being Element of holds
( p * e = p & e * p = p & ex g being Element of st
( p * g = e & g * p = e ) )
reconsider e' = idseq n as Element of Permutations n by Def11;
A2: for p being Element of ex g being Element of st
( p * g = e & g * p = e )

proof

let p be Element of ; :: thesis: ex g being Element of st
( p * g = e & g * p = e )
reconsider q = p as Element of Permutations n by Def13;
reconsider r = q as Permutation of Seg n by Def11;
reconsider f = r " as Element of Permutations n by Def11;
reconsider g = f as Element of by Th26;
take g ; :: thesis: ( p * g = e & g * p = e )
A3: g * p = q * f by Def13
.= e by Th25 ;
p * g = f * q by Def13
.= e by Th25 ;
hence ( p * g = e & g * p = e ) by A3; :: thesis: verum

end;

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

proof

let p be Element of ; :: thesis: ( p * e = p & e * p = p )
reconsider p' = p as Element of Permutations n by Def13;
A4: e * p = p' * e' by Def13
.= p by Th24 ;
p * e = e' * p' by Def13
.= p by Th24 ;
hence ( p * e = p & e * p = p ) by A4; :: thesis: verum

end;

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

end;

for p, q, r being Element of holds (p * q) * r = p * (q * r)

proof

let p, q, r be Element of ; :: thesis: (p * q) * r = p * (q * r)
reconsider p' = p, q' = q, r' = r as Element of Permutations n by Def13;
A5: ( q' * p' is Permutation of Seg n & r' * q' is Permutation of Seg n )

proof

reconsider p' = p', q' = q', r' = r' as Permutation of Seg n by Def11;
( q' * p' is Permutation of Seg n & r' * q' is Permutation of Seg n ) ;
hence ( q' * p' is Permutation of Seg n & r' * q' is Permutation of Seg n ) ; :: thesis: verum

end;

then A6: q' * p' is Element of Permutations n by Def11;
A7: r' * q' is Element of Permutations n by A5, Def11;
(p * q) * r = the multF of (Group_of_Perm n) . (q' * p'),r' by Def13
.= r' * (q' * p') by A6, Def13
.= (r' * q') * p' by RELAT_1:55
.= the multF of (Group_of_Perm n) . p',(r' * q') by A7, Def13
.= p * (q * r) by Def13 ;
hence (p * q) * r = p * (q * r) ; :: thesis: verum

end;

hence ( Group_of_Perm n is associative & Group_of_Perm n is Group-like ) by A1, GROUP_1:5; :: thesis: verum