let I be non empty set ; :: thesis: for J being non-Empty TopSpace-yielding ManySortedSet of I
for P being non empty Subset of (product ()) st P in product_prebasis J holds
( ( for j being Element of I holds (proj (J,j)) .: P is open ) & ex i being Element of I st
for j being Element of I st j <> i holds
(proj (J,j)) .: P = [#] (J . j) )

let J be non-Empty TopSpace-yielding ManySortedSet of I; :: thesis: for P being non empty Subset of (product ()) st P in product_prebasis J holds
( ( for j being Element of I holds (proj (J,j)) .: P is open ) & ex i being Element of I st
for j being Element of I st j <> i holds
(proj (J,j)) .: P = [#] (J . j) )

let P be non empty Subset of (product ()); :: thesis: ( P in product_prebasis J implies ( ( for j being Element of I holds (proj (J,j)) .: P is open ) & ex i being Element of I st
for j being Element of I st j <> i holds
(proj (J,j)) .: P = [#] (J . j) ) )

assume P in product_prebasis J ; :: thesis: ( ( for j being Element of I holds (proj (J,j)) .: P is open ) & ex i being Element of I st
for j being Element of I st j <> i holds
(proj (J,j)) .: P = [#] (J . j) )

then consider i being Element of I such that
A1: (proj (J,i)) .: P is open and
A2: for j being Element of I st j <> i holds
(proj (J,j)) .: P = [#] (J . j) by Th57;
hereby :: thesis: ex i being Element of I st
for j being Element of I st j <> i holds
(proj (J,j)) .: P = [#] (J . j)
let j be Element of I; :: thesis: (proj (J,j)) .: P is open
( j <> i implies (proj (J,j)) .: P = [#] (J . j) ) by A2;
hence (proj (J,j)) .: P is open by A1; :: thesis: verum
end;
take i ; :: thesis: for j being Element of I st j <> i holds
(proj (J,j)) .: P = [#] (J . j)

thus for j being Element of I st j <> i holds
(proj (J,j)) .: P = [#] (J . j) by A2; :: thesis: verum