let f1, f2 be QuaOp of F₁(); :: thesis: ( ( for a, b, c, d being Element of F₁() holds f1 . (a,b,c,d) = F₂(a,b,c,d) ) & ( for a, b, c, d being Element of F₁() holds f2 . (a,b,c,d) = F₂(a,b,c,d) ) implies f1 = f2 )
assume that
A1: for a, b, c, d being Element of F₁() holds f1 . (a,b,c,d) = F₂(a,b,c,d) and
A2: for a, b, c, d being Element of F₁() holds f2 . (a,b,c,d) = F₂(a,b,c,d) ; :: thesis: f1 = f2

now :: thesis: for a, b, c, d being Element of F₁() holds f1 . (a,b,c,d) = f2 . (a,b,c,d)

let a, b, c, d be Element of F₁(); :: thesis: f1 . (a,b,c,d) = f2 . (a,b,c,d)
thus f1 . (a,b,c,d) = F₂(a,b,c,d) by A1
.= f2 . (a,b,c,d) by A2 ; :: thesis: verum

end;

hence f1 = f2 by MULTOP_1:6; :: thesis: verum