let A, B be ManySortedSet of the carrier of S; :: thesis: ( ( for s being SortSymbol of S holds A . s = FreeSort (X,s) ) & ( for s being SortSymbol of S holds B . s = FreeSort (X,s) ) implies A = B )
assume that
A2: for s being SortSymbol of S holds A . s = FreeSort (X,s) and
A3: for s being SortSymbol of S holds B . s = FreeSort (X,s) ; :: thesis: A = B
for i being object st i in the carrier of S holds
A . i = B . i

proof

let i be object ; :: thesis: ( i in the carrier of S implies A . i = B . i )
assume i in the carrier of S ; :: thesis: A . i = B . i
then reconsider s = i as SortSymbol of S ;
A . s = FreeSort (X,s) by A2;
hence A . i = B . i by A3; :: thesis: verum

end;

hence A = B ; :: thesis: verum