let X be non empty TopSpace; :: thesis: for X1, X2, Y1, Y2 being SubSpace of X st X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition & X1,X2 are_weakly_separated holds

Y1,Y2 are_weakly_separated

let X1, X2, Y1, Y2 be SubSpace of X; :: thesis: ( X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition & X1,X2 are_weakly_separated implies Y1,Y2 are_weakly_separated )

assume A1: for A1, B1 being Subset of X st A1 = the carrier of X1 & B1 = the carrier of Y1 holds

A1,B1 constitute_a_decomposition ; :: according to TSEP_2:def 2 :: thesis: ( not X2,Y2 constitute_a_decomposition or not X1,X2 are_weakly_separated or Y1,Y2 are_weakly_separated )

assume A2: for A2, B2 being Subset of X st A2 = the carrier of X2 & B2 = the carrier of Y2 holds

A2,B2 constitute_a_decomposition ; :: according to TSEP_2:def 2 :: thesis: ( not X1,X2 are_weakly_separated or Y1,Y2 are_weakly_separated )

assume A3: for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds

A1,A2 are_weakly_separated ; :: according to TSEP_1:def 7 :: thesis: Y1,Y2 are_weakly_separated

Y1,Y2 are_weakly_separated

let X1, X2, Y1, Y2 be SubSpace of X; :: thesis: ( X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition & X1,X2 are_weakly_separated implies Y1,Y2 are_weakly_separated )

assume A1: for A1, B1 being Subset of X st A1 = the carrier of X1 & B1 = the carrier of Y1 holds

A1,B1 constitute_a_decomposition ; :: according to TSEP_2:def 2 :: thesis: ( not X2,Y2 constitute_a_decomposition or not X1,X2 are_weakly_separated or Y1,Y2 are_weakly_separated )

assume A2: for A2, B2 being Subset of X st A2 = the carrier of X2 & B2 = the carrier of Y2 holds

A2,B2 constitute_a_decomposition ; :: according to TSEP_2:def 2 :: thesis: ( not X1,X2 are_weakly_separated or Y1,Y2 are_weakly_separated )

assume A3: for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds

A1,A2 are_weakly_separated ; :: according to TSEP_1:def 7 :: thesis: Y1,Y2 are_weakly_separated

now :: thesis: for B1, B2 being Subset of X st B1 = the carrier of Y1 & B2 = the carrier of Y2 holds

B1,B2 are_weakly_separated

hence
Y1,Y2 are_weakly_separated
; :: thesis: verumB1,B2 are_weakly_separated

reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by TSEP_1:1;

let B1, B2 be Subset of X; :: thesis: ( B1 = the carrier of Y1 & B2 = the carrier of Y2 implies B1,B2 are_weakly_separated )

assume ( B1 = the carrier of Y1 & B2 = the carrier of Y2 ) ; :: thesis: B1,B2 are_weakly_separated

then A4: ( A1,B1 constitute_a_decomposition & A2,B2 constitute_a_decomposition ) by A1, A2;

A1,A2 are_weakly_separated by A3;

hence B1,B2 are_weakly_separated by A4, Th15; :: thesis: verum

end;let B1, B2 be Subset of X; :: thesis: ( B1 = the carrier of Y1 & B2 = the carrier of Y2 implies B1,B2 are_weakly_separated )

assume ( B1 = the carrier of Y1 & B2 = the carrier of Y2 ) ; :: thesis: B1,B2 are_weakly_separated

then A4: ( A1,B1 constitute_a_decomposition & A2,B2 constitute_a_decomposition ) by A1, A2;

A1,A2 are_weakly_separated by A3;

hence B1,B2 are_weakly_separated by A4, Th15; :: thesis: verum