let GX be non empty TopSpace; :: thesis: for A, B being Subset of GX st A is connected & B is connected & A <> {} & A c= B holds

B c= Component_of A

let A, B be Subset of GX; :: thesis: ( A is connected & B is connected & A <> {} & A c= B implies B c= Component_of A )

assume that

A1: A is connected and

A2: B is connected and

A3: ( A <> {} & A c= B ) ; :: thesis: B c= Component_of A

Component_of A = Component_of B by A1, A2, A3, Th12;

hence B c= Component_of A by A2, Th1; :: thesis: verum

B c= Component_of A

let A, B be Subset of GX; :: thesis: ( A is connected & B is connected & A <> {} & A c= B implies B c= Component_of A )

assume that

A1: A is connected and

A2: B is connected and

A3: ( A <> {} & A c= B ) ; :: thesis: B c= Component_of A

Component_of A = Component_of B by A1, A2, A3, Th12;

hence B c= Component_of A by A2, Th1; :: thesis: verum