let X be non empty TopSpace; :: thesis: for X0, X1, X2 being non empty SubSpace of X st X1 meets X0 & X2 meets X0 holds

for Y1, Y2 being SubSpace of X0 st Y1 = X1 meet X0 & Y2 = X2 meet X0 & X1,X2 are_weakly_separated holds

Y1,Y2 are_weakly_separated

let X0 be non empty SubSpace of X; :: thesis: for X1, X2 being non empty SubSpace of X st X1 meets X0 & X2 meets X0 holds

for Y1, Y2 being SubSpace of X0 st Y1 = X1 meet X0 & Y2 = X2 meet X0 & X1,X2 are_weakly_separated holds

Y1,Y2 are_weakly_separated

let X1, X2 be non empty SubSpace of X; :: thesis: ( X1 meets X0 & X2 meets X0 implies for Y1, Y2 being SubSpace of X0 st Y1 = X1 meet X0 & Y2 = X2 meet X0 & X1,X2 are_weakly_separated holds

Y1,Y2 are_weakly_separated )

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

assume A1: ( X1 meets X0 & X2 meets X0 ) ; :: thesis: for Y1, Y2 being SubSpace of X0 st Y1 = X1 meet X0 & Y2 = X2 meet X0 & X1,X2 are_weakly_separated holds

Y1,Y2 are_weakly_separated

let Y1, Y2 be SubSpace of X0; :: thesis: ( Y1 = X1 meet X0 & Y2 = X2 meet X0 & X1,X2 are_weakly_separated implies Y1,Y2 are_weakly_separated )

assume A2: ( Y1 = X1 meet X0 & Y2 = X2 meet X0 ) ; :: thesis: ( not X1,X2 are_weakly_separated or Y1,Y2 are_weakly_separated )

assume X1,X2 are_weakly_separated ; :: thesis: Y1,Y2 are_weakly_separated

then A3: A1,A2 are_weakly_separated ;

for Y1, Y2 being SubSpace of X0 st Y1 = X1 meet X0 & Y2 = X2 meet X0 & X1,X2 are_weakly_separated holds

Y1,Y2 are_weakly_separated

let X0 be non empty SubSpace of X; :: thesis: for X1, X2 being non empty SubSpace of X st X1 meets X0 & X2 meets X0 holds

for Y1, Y2 being SubSpace of X0 st Y1 = X1 meet X0 & Y2 = X2 meet X0 & X1,X2 are_weakly_separated holds

Y1,Y2 are_weakly_separated

let X1, X2 be non empty SubSpace of X; :: thesis: ( X1 meets X0 & X2 meets X0 implies for Y1, Y2 being SubSpace of X0 st Y1 = X1 meet X0 & Y2 = X2 meet X0 & X1,X2 are_weakly_separated holds

Y1,Y2 are_weakly_separated )

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

assume A1: ( X1 meets X0 & X2 meets X0 ) ; :: thesis: for Y1, Y2 being SubSpace of X0 st Y1 = X1 meet X0 & Y2 = X2 meet X0 & X1,X2 are_weakly_separated holds

Y1,Y2 are_weakly_separated

let Y1, Y2 be SubSpace of X0; :: thesis: ( Y1 = X1 meet X0 & Y2 = X2 meet X0 & X1,X2 are_weakly_separated implies Y1,Y2 are_weakly_separated )

assume A2: ( Y1 = X1 meet X0 & Y2 = X2 meet X0 ) ; :: thesis: ( not X1,X2 are_weakly_separated or Y1,Y2 are_weakly_separated )

assume X1,X2 are_weakly_separated ; :: thesis: Y1,Y2 are_weakly_separated

then A3: A1,A2 are_weakly_separated ;

now :: thesis: for C1, C2 being Subset of X0 st C1 = the carrier of Y1 & C2 = the carrier of Y2 holds

C1,C2 are_weakly_separated

hence
Y1,Y2 are_weakly_separated
; :: thesis: verumC1,C2 are_weakly_separated

let C1, C2 be Subset of X0; :: thesis: ( C1 = the carrier of Y1 & C2 = the carrier of Y2 implies C1,C2 are_weakly_separated )

assume ( C1 = the carrier of Y1 & C2 = the carrier of Y2 ) ; :: thesis: C1,C2 are_weakly_separated

then ( C1 = the carrier of X0 /\ A1 & C2 = the carrier of X0 /\ A2 ) by A1, A2, TSEP_1:def 4;

hence C1,C2 are_weakly_separated by A3, Th28; :: thesis: verum

end;assume ( C1 = the carrier of Y1 & C2 = the carrier of Y2 ) ; :: thesis: C1,C2 are_weakly_separated

then ( C1 = the carrier of X0 /\ A1 & C2 = the carrier of X0 /\ A2 ) by A1, A2, TSEP_1:def 4;

hence C1,C2 are_weakly_separated by A3, Th28; :: thesis: verum