let X be non empty TopSpace; 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_separated holds
Y1,Y2 are_separated
let X0 be non empty SubSpace of X; 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_separated holds
Y1,Y2 are_separated
let X1, X2 be non empty SubSpace of X; ( 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_separated holds
Y1,Y2 are_separated )
assume A1:
( X1 meets X0 & X2 meets X0 )
; for Y1, Y2 being SubSpace of X0 st Y1 = X1 meet X0 & Y2 = X2 meet X0 & X1,X2 are_separated holds
Y1,Y2 are_separated
let Y1, Y2 be SubSpace of X0; ( Y1 = X1 meet X0 & Y2 = X2 meet X0 & X1,X2 are_separated implies Y1,Y2 are_separated )
assume A2:
( Y1 = X1 meet X0 & Y2 = X2 meet X0 )
; ( not X1,X2 are_separated or Y1,Y2 are_separated )
assume
X1,X2 are_separated
; Y1,Y2 are_separated
then
X1 meet X0,X2 meet X0 are_separated
by A1, TSEP_1:70;
hence
Y1,Y2 are_separated
by A2, Th44; verum