let D be non empty set ; :: thesis: for d1, d2 being Element of D
for g being BinOp of D st g is commutative holds
g "**" <*d1,d2*> = g "**" <*d2,d1*>
let d1, d2 be Element of D; :: thesis: for g being BinOp of D st g is commutative holds
g "**" <*d1,d2*> = g "**" <*d2,d1*>
let g be BinOp of D; :: thesis: ( g is commutative implies g "**" <*d1,d2*> = g "**" <*d2,d1*> )
assume A1: g is commutative ; :: thesis: g "**" <*d1,d2*> = g "**" <*d2,d1*>
thus g "**" <*d1,d2*> = g . (d1,d2) by Th12
.= g . (d2,d1) by A1, BINOP_1:def 2
.= g "**" <*d2,d1*> by Th12 ; :: thesis: verum