let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X
for x being Point of (X1 union X2)
for U1 being Subset of X1
for U2 being Subset of X2 st U1 is open & x in U1 & U2 is open & x in U2 holds
ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= U1 \/ U2 )
let X1, X2 be non empty SubSpace of X; :: thesis: for x being Point of (X1 union X2)
for U1 being Subset of X1
for U2 being Subset of X2 st U1 is open & x in U1 & U2 is open & x in U2 holds
ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= U1 \/ U2 )
let x be Point of (X1 union X2); :: thesis: for U1 being Subset of X1
for U2 being Subset of X2 st U1 is open & x in U1 & U2 is open & x in U2 holds
ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= U1 \/ U2 )
let U1 be Subset of X1; :: thesis: for U2 being Subset of X2 st U1 is open & x in U1 & U2 is open & x in U2 holds
ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= U1 \/ U2 )
let U2 be Subset of X2; :: thesis: ( U1 is open & x in U1 & U2 is open & x in U2 implies ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= U1 \/ U2 ) )
assume that
A1: ( U1 is open & x in U1 ) and
A2: ( U2 is open & x in U2 ) ; :: thesis: ex V being Subset of (X1 union X2) st
( V is open & x in V & V c= U1 \/ U2 )
A3: X1 is SubSpace of X1 union X2 by TSEP_1:22;
then consider V1 being Subset of (X1 union X2) such that
A4: ( V1 is open & V1 /\ ([#] X1) = U1 ) by A1, TOPS_2:32;
A5: X2 is SubSpace of X1 union X2 by TSEP_1:22;
then consider V2 being Subset of (X1 union X2) such that
A6: ( V2 is open & V2 /\ ([#] X2) = U2 ) by A2, TOPS_2:32;
take V = V1 /\ V2; :: thesis: ( V is open & x in V & V c= U1 \/ U2 )
reconsider C1 = the carrier of X1 as Subset of (X1 union X2) by A3, TSEP_1:1;
reconsider C2 = the carrier of X2 as Subset of (X1 union X2) by A5, TSEP_1:1;
the carrier of (X1 union X2) = C1 \/ C2 by TSEP_1:def 2;
then A7: V = V /\ (C1 \/ C2) by XBOOLE_1:28
.= (V /\ C1) \/ (V /\ C2) by XBOOLE_1:23 ;
A8: ( V /\ C1 c= V1 /\ C1 & V /\ C2 c= V2 /\ C2 ) by XBOOLE_1:17, XBOOLE_1:26;
( x in V1 & x in V2 ) by A1, A2, A4, A6, XBOOLE_0:def 4;
hence ( V is open & x in V & V c= U1 \/ U2 ) by A4, A6, A7, A8, TOPS_1:38, XBOOLE_0:def 4, XBOOLE_1:13; :: thesis: verum