let X be non empty TopSpace; :: thesis: for X1, X2, Y1, Y2 being non empty SubSpace of X st X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition & X1,X2 are_separated holds
Y1,Y2 are_weakly_separated
let X1, X2, Y1, Y2 be non empty SubSpace of X; :: thesis: ( X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition & X1,X2 are_separated implies Y1,Y2 are_weakly_separated )
assume A1: ( X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition ) ; :: thesis: ( not X1,X2 are_separated or Y1,Y2 are_weakly_separated )
assume X1,X2 are_separated ; :: thesis: Y1,Y2 are_weakly_separated
then X1,X2 are_weakly_separated by TSEP_1:78;
hence Y1,Y2 are_weakly_separated by A1, Th39; :: thesis: verum