let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2) holds
( ex x1 being Point of X1 st x1 = x or ex x2 being Point of X2 st x2 = x )
let X1, X2 be non empty SubSpace of X; :: thesis: for x being Point of (X1 union X2) holds
( ex x1 being Point of X1 st x1 = x or ex x2 being Point of X2 st x2 = x )
let x be Point of (X1 union X2); :: thesis: ( ex x1 being Point of X1 st x1 = x or ex x2 being Point of X2 st x2 = x )
reconsider A0 = the carrier of (X1 union X2) as Subset of X by TSEP_1:1;
reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
assume A1: for x1 being Point of X1 holds not x1 = x ; :: thesis: ex x2 being Point of X2 st x2 = x
ex x2 being Point of X2 st x2 = x

proof

( A0 = A1 \/ A2 & not x in A1 ) by A1, TSEP_1:def 2;
then reconsider x2 = x as Point of X2 by XBOOLE_0:def 3;
take x2 ; :: thesis: x2 = x
thus x2 = x ; :: thesis: verum

end;

hence ex x2 being Point of X2 st x2 = x ; :: thesis: verum