let G be Group; :: thesis: for a, b being Element of G holds [.a,b.] " = [.a,(b ").] |^ b
let a, b be Element of G; :: thesis: [.a,b.] " = [.a,(b ").] |^ b
thus [.a,b.] " = (((a ") * (b ")) * (a * b)) " by GROUP_1:def 3
.= ((a * b) ") * (((a ") * (b ")) ") by GROUP_1:17
.= ((b ") * (a ")) * (((a ") * (b ")) ") by GROUP_1:17
.= ((b ") * (a ")) * (((b ") ") * ((a ") ")) by GROUP_1:17
.= (b ") * ((a ") * (b * a)) by GROUP_1:def 3
.= (b ") * (((a ") * b) * a) by GROUP_1:def 3
.= ((b ") * (((a ") * b) * a)) * (1_ G) by GROUP_1:def 4
.= ((b ") * (((a ") * b) * a)) * ((b ") * b) by GROUP_1:def 5
.= (((b ") * (((a ") * b) * a)) * (b ")) * b by GROUP_1:def 3
.= [.a,(b ").] |^ b by GROUP_1:def 3 ; :: thesis: verum