reconsider e = id the carrier of G as Element of multMagma(# (Aut G),(AutComp G) #) by Th3;
take e ; :: thesis: for h being Element of multMagma(# (Aut G),(AutComp G) #) holds
( h * e = h & e * h = h & ex g being Element of multMagma(# (Aut G),(AutComp G) #) st
( h * g = e & g * h = e ) )
let h be Element of multMagma(# (Aut G),(AutComp G) #); :: thesis: ( h * e = h & e * h = h & ex g being Element of multMagma(# (Aut G),(AutComp G) #) st
( h * g = e & g * h = e ) )
consider A being Element of Aut G such that
A2: A = h ;
h * e = A * (id the carrier of G) by A2, Def2
.= A by FUNCT_2:17 ;
hence h * e = h by A2; :: thesis: ( e * h = h & ex g being Element of multMagma(# (Aut G),(AutComp G) #) st
( h * g = e & g * h = e ) )
e * h = (id the carrier of G) * A by A2, Def2
.= A by FUNCT_2:17 ;
hence e * h = h by A2; :: thesis: ex g being Element of multMagma(# (Aut G),(AutComp G) #) st
( h * g = e & g * h = e )
reconsider g = A " as Element of multMagma(# (Aut G),(AutComp G) #) by Th6;
take g ; :: thesis: ( h * g = e & g * h = e )
reconsider A = A as Homomorphism of G,G by Def1;
A3: A is one-to-one by Def1;
A is onto by Def1;
then A4: rng A = the carrier of G ;
thus h * g = A * (A ") by A2, Def2
.= e by A3, A4, FUNCT_2:29 ; :: thesis: g * h = e
thus g * h = (A ") * A by A2, Def2
.= e by A3, A4, FUNCT_2:29 ; :: thesis: verum

let f, g, h be Element of multMagma(# (Aut G),(AutComp G) #); :: thesis: (f * g) * h = f * (g * h)
reconsider A = f, B = g, C = h as Element of Aut G ;
A5: g * h = B * C by Def2;
f * g = A * B by Def2;
hence (f * g) * h = (A * B) * C by Def2
.= A * (B * C) by RELAT_1:36
.= f * (g * h) by A5, Def2 ;
:: thesis: verum