let o1, o2 be BinOp of F₁(); :: thesis: ( ( for a, b being Element of F₁() holds o1 . a,b = F₂(a,b) ) & ( for a, b being Element of F₁() holds o2 . a,b = F₂(a,b) ) implies o1 = o2 )
assume that
A1: for a, b being Element of F₁() holds o1 . a,b = F₂(a,b) and
A2: for a, b being Element of F₁() holds o2 . a,b = F₂(a,b) ; :: thesis: o1 = o2

now

let a, b be Element of F₁(); :: thesis: o1 . a,b = o2 . a,b
thus o1 . a,b = F₂(a,b) by A1
.= o2 . a,b by A2 ; :: thesis: verum

end;

hence o1 = o2 by BINOP_1:2; :: thesis: verum