let d, d9 be BinOp of (product a); :: thesis: ( ( for f, g being Element of product a
for i being Element of dom a holds (d . (f,g)) . i = (b . i) . ((f . i),(g . i)) ) & ( for f, g being Element of product a
for i being Element of dom a holds (d9 . (f,g)) . i = (b . i) . ((f . i),(g . i)) ) implies d = d9 )
assume that
A5: for f, g being Element of product a
for i being Element of dom a holds (d . (f,g)) . i = (b . i) . ((f . i),(g . i)) and
A6: for f, g being Element of product a
for i being Element of dom a holds (d9 . (f,g)) . i = (b . i) . ((f . i),(g . i)) ; :: thesis: d = d9
now :: thesis: for f, g being Element of product a
for i being Element of dom a holds (d . (f,g)) . i = (d9 . (f,g)) . i
let f, g be Element of product a; :: thesis: for i being Element of dom a holds (d . (f,g)) . i = (d9 . (f,g)) . i
let i be Element of dom a; :: thesis: (d . (f,g)) . i = (d9 . (f,g)) . i
thus (d . (f,g)) . i = (b . i) . ((f . i),(g . i)) by A5
.= (d9 . (f,g)) . i by A6 ; :: thesis: verum
end;
hence d = d9 by Th16; :: thesis: verum