let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X st X1 union X2 is closed SubSpace of X & X1,X2 are_separated holds
X1 is closed SubSpace of X
let X1, X2 be non empty SubSpace of X; :: thesis: ( X1 union X2 is closed SubSpace of X & 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 A1: X1 union X2 is closed SubSpace of X ; :: thesis: ( not X1,X2 are_separated or X1 is closed SubSpace of X )
assume X1,X2 are_separated ; :: thesis: X1 is closed SubSpace of X
then A2: A1,A2 are_separated by Def8;
A1 \/ A2 = the carrier of (X1 union X2) by Def2;
then A1 \/ A2 is closed by A1, Th11;
then A1 is closed by A2, Th39;
hence X1 is closed SubSpace of X by Th11; :: thesis: verum