let GX be TopSpace; :: thesis: for A, B being Subset of GX st A is a_component & B is a_component & not A = B holds
A,B are_separated
let A, B be Subset of GX; :: thesis: ( A is a_component & B is a_component & not A = B implies A,B are_separated )
assume that
A1: A is a_component and
A2: B is a_component ; :: thesis: ( A = B or A,B are_separated )
( A <> B implies A,B are_separated )
proof
A3: B c= A \/ B by XBOOLE_1:7;
A4: A c= A \/ B by XBOOLE_1:7;
assume A5: A <> B ; :: thesis: A,B are_separated
assume not A,B are_separated ; :: thesis: contradiction
then A \/ B is connected by A1, A2, Th17;
then A = A \/ B by A1, A4;
hence contradiction by A1, A2, A5, A3; :: thesis: verum
end;
hence ( A = B or A,B are_separated ) ; :: thesis: verum