let I be set ; :: thesis: for x, A being ManySortedSet of st x in {A} holds
x = A
let x, A be ManySortedSet of ; :: thesis: ( x in {A} implies x = A )
assume A1: x in {A} ; :: thesis: x = A

now

let i be set ; :: thesis: ( i in I implies x . i = A . i )
assume A2: i in I ; :: thesis: x . i = A . i
then x . i in {A} . i by A1, PBOOLE:def 4;
then x . i in {(A . i)} by A2, Def1;
hence x . i = A . i by TARSKI:def 1; :: thesis: verum

end;

hence x = A by PBOOLE:3; :: thesis: verum