let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1,X2 are_separated holds
X1 is closed SubSpace of X
let X1, X2 be non empty SubSpace of X; :: thesis: ( X = X1 union X2 & X1,X2 are_separated implies X1 is closed SubSpace of X )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
assume X = X1 union X2 ; :: thesis: ( not X1,X2 are_separated or X1 is closed SubSpace of X )
then A1: A1 \/ A2 = [#] X by Def2;
assume X1,X2 are_separated ; :: thesis: X1 is closed SubSpace of X
then A1,A2 are_separated ;
then A1 is closed by A1, CONNSP_1:4;
hence X1 is closed SubSpace of X by Th11; :: thesis: verum