let T be non empty TopSpace; :: thesis: for a, b being Element of (Closed_Domains_Lattice T)
for A, B being Element of Closed_Domains_of T st a = A & b = B holds
( a [= b iff A c= B )
let a, b be Element of (Closed_Domains_Lattice T); :: thesis: for A, B being Element of Closed_Domains_of T st a = A & b = B holds
( a [= b iff A c= B )
let A, B be Element of Closed_Domains_of T; :: thesis: ( a = A & b = B implies ( a [= b iff A c= B ) )
assume that
A1: a = A and
A2: b = B ; :: thesis: ( a [= b iff A c= B )
thus ( a [= b implies A c= B ) :: thesis: ( A c= B implies a [= b )
proof
assume a [= b ; :: thesis: A c= B
then a "\/" b = b by LATTICES:def 3;
then A \/ B = B by A1, A2, Th94;
hence A c= B by XBOOLE_1:7; :: thesis: verum
end;
thus ( A c= B implies a [= b ) :: thesis: verum
proof
assume A c= B ; :: thesis: a [= b
then A \/ B = B by XBOOLE_1:12;
then a "\/" b = b by A1, A2, Th94;
hence a [= b by LATTICES:def 3; :: thesis: verum
end;