let I be set ; :: thesis: for x, A being ManySortedSet of I st x in {A} holds
x = A
let x, A be ManySortedSet of I; :: thesis: ( x in {A} implies x = A )
assume A1: x in {A} ; :: thesis: x = A
now :: thesis: for i being object st i in I holds
x . i = A . i
let i be object ; :: 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;
then x . i in {(A . i)} by A2, Def1;
hence x . i = A . i by TARSKI:def 1; :: thesis: verum
end;
hence x = A ; :: thesis: verum