let X, Y be ManySortedSet of I; :: thesis: ( ( for i being set st i in I holds
X . i = {} (A . i) ) & ( for i being set st i in I holds
Y . i = {} (A . i) ) implies X = Y )
assume that
A2: for i being set st i in I holds
X . i = {} (A . i) and
A3: for i being set st i in I holds
Y . i = {} (A . i) ; :: thesis: X = Y
for i being set st i in I holds
X . i = Y . i

proof

let i be set ; :: thesis: ( i in I implies X . i = Y . i )
assume A4: i in I ; :: thesis: X . i = Y . i
then X . i = {} (A . i) by A2;
hence X . i = Y . i by A3, A4; :: thesis: verum

end;

hence X = Y by PBOOLE:3; :: thesis: verum