let f, g be Function of ( the Sorts of U1 . s),((Class R) . s); :: thesis: ( ( for x being set st x in the Sorts of U1 . s holds
f . x = Class ((R . s),x) ) & ( for x being set st x in the Sorts of U1 . s holds
g . x = Class ((R . s),x) ) implies f = g )
assume that
A3: for x being set st x in the Sorts of U1 . s holds
f . x = Class ((R . s),x) and
A4: for x being set st x in the Sorts of U1 . s holds
g . x = Class ((R . s),x) ; :: thesis: f = g

A5: now :: thesis: for x being object st x in the Sorts of U1 . s holds
f . x = g . x

let x be object ; :: thesis: ( x in the Sorts of U1 . s implies f . x = g . x )
assume A6: x in the Sorts of U1 . s ; :: thesis: f . x = g . x
then f . x = Class ((R . s),x) by A3;
hence f . x = g . x by A4, A6; :: thesis: verum

end;

( dom f = the Sorts of U1 . s & dom g = the Sorts of U1 . s ) by FUNCT_2:def 1;
hence f = g by A5, FUNCT_1:2; :: thesis: verum