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_separated holds

Y1,Y2 are_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_separated holds

Y1,Y2 are_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_separated holds

Y1,Y2 are_separated )

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_separated holds

Y1,Y2 are_separated

let Y1, Y2 be SubSpace of X0; :: thesis: ( 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 ) ; :: thesis: ( not X1,X2 are_separated or Y1,Y2 are_separated )

assume X1,X2 are_separated ; :: thesis: 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; :: thesis: verum

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; :: 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_separated holds

Y1,Y2 are_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_separated holds

Y1,Y2 are_separated )

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_separated holds

Y1,Y2 are_separated

let Y1, Y2 be SubSpace of X0; :: thesis: ( 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 ) ; :: thesis: ( not X1,X2 are_separated or Y1,Y2 are_separated )

assume X1,X2 are_separated ; :: thesis: 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; :: thesis: verum