let X be non empty TopSpace; :: thesis: for A1, A, A2 being Subset of X st A1,A constitute_a_decomposition & A,A2 constitute_a_decomposition holds
A1 = A2
let A1, A, A2 be Subset of X; :: thesis: ( A1,A constitute_a_decomposition & A,A2 constitute_a_decomposition implies A1 = A2 )
assume A1,A constitute_a_decomposition ; :: thesis: ( not A,A2 constitute_a_decomposition or A1 = A2 )
then A1: A1 = A ` by Th4;
assume A,A2 constitute_a_decomposition ; :: thesis: A1 = A2
hence A1 = A2 by A1, Th4; :: thesis: verum