let A1, A2 be set ; :: thesis: ( ( for x being set holds
( x in A1 iff ex M, N being strict feasible MSAlgebra over S ex f being ManySortedFunction of M,N st
( M = i & N = j & f = x & the Sorts of M is_transformable_to the Sorts of N & f is_homomorphism M,N ) ) ) & ( for x being set holds
( x in A2 iff ex M, N being strict feasible MSAlgebra over S ex f being ManySortedFunction of M,N st
( M = i & N = j & f = x & the Sorts of M is_transformable_to the Sorts of N & f is_homomorphism M,N ) ) ) implies A1 = A2 )
assume A22: for x being set holds
( x in A1 iff ex M, N being strict feasible MSAlgebra over S ex f being ManySortedFunction of M,N st
( M = i & N = j & f = x & the Sorts of M is_transformable_to the Sorts of N & f is_homomorphism M,N ) ) ; :: thesis: ( ex x being set st
( ( x in A2 implies ex M, N being strict feasible MSAlgebra over S ex f being ManySortedFunction of M,N st
( M = i & N = j & f = x & the Sorts of M is_transformable_to the Sorts of N & f is_homomorphism M,N ) ) implies ( ex M, N being strict feasible MSAlgebra over S ex f being ManySortedFunction of M,N st
( M = i & N = j & f = x & the Sorts of M is_transformable_to the Sorts of N & f is_homomorphism M,N ) & not x in A2 ) ) or A1 = A2 )
assume A23: for x being set holds
( x in A2 iff ex M, N being strict feasible MSAlgebra over S ex f being ManySortedFunction of M,N st
( M = i & N = j & f = x & the Sorts of M is_transformable_to the Sorts of N & f is_homomorphism M,N ) ) ; :: thesis: A1 = A2
A24: A2 c= A1
proof
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in A2 or a in A1 )
assume a in A2 ; :: thesis: a in A1
then ex M, N being strict feasible MSAlgebra over S ex f being ManySortedFunction of M,N st
( M = i & N = j & f = a & the Sorts of M is_transformable_to the Sorts of N & f is_homomorphism M,N ) by A23;
hence a in A1 by A22; :: thesis: verum
end;
A1 c= A2
proof
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in A1 or a in A2 )
assume a in A1 ; :: thesis: a in A2
then ex M, N being strict feasible MSAlgebra over S ex f being ManySortedFunction of M,N st
( M = i & N = j & f = a & the Sorts of M is_transformable_to the Sorts of N & f is_homomorphism M,N ) by A22;
hence a in A2 by A23; :: thesis: verum
end;
hence A1 = A2 by A24, XBOOLE_0:def 10; :: thesis: verum