let G be Group; :: thesis: for a, b being Element of G holds [.a,b.] " = [.b,a.]
let a, b be Element of G; :: thesis: [.a,b.] " = [.b,a.]
thus [.a,b.] " = (((a " ) * (b " )) * (a * b)) " by Th19
.= ((a * b) " ) * (((a " ) * (b " )) " ) by GROUP_1:25
.= ((b " ) * (a " )) * (((a " ) * (b " )) " ) by GROUP_1:25
.= ((b " ) * (a " )) * (((b " ) " ) * ((a " ) " )) by GROUP_1:25
.= ((b " ) * (a " )) * (((b " ) " ) * a)
.= ((b " ) * (a " )) * (b * a)
.= [.b,a.] by Th19 ; :: thesis: verum