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

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

hence
ex x2 being Point of X2 st x2 = x
; :: thesis: verum
( 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;then reconsider x2 = x as Point of X2 by XBOOLE_0:def 3;

take x2 ; :: thesis: x2 = x

thus x2 = x ; :: thesis: verum