let X be non empty TopSpace; for A1, A2, C1, C2 being Subset of X st A1,C1 constitute_a_decomposition & A2,C2 constitute_a_decomposition & A1 misses A2 & C1,C2 are_weakly_separated holds
A1,A2 are_separated
let A1, A2, C1, C2 be Subset of X; ( A1,C1 constitute_a_decomposition & A2,C2 constitute_a_decomposition & A1 misses A2 & C1,C2 are_weakly_separated implies A1,A2 are_separated )
assume A1:
( A1,C1 constitute_a_decomposition & A2,C2 constitute_a_decomposition )
; ( not A1 misses A2 or not C1,C2 are_weakly_separated or A1,A2 are_separated )
assume A2:
A1 /\ A2 = {}
; XBOOLE_0:def 7 ( not C1,C2 are_weakly_separated or A1,A2 are_separated )
assume
C1,C2 are_weakly_separated
; A1,A2 are_separated
then A3:
A1,A2 are_weakly_separated
by A1, Th15;
A1 misses A2
by A2;
hence
A1,A2 are_separated
by A3, TSEP_1:46; verum