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 F1 being Subset of X1

for F2 being Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 holds

ex H being Subset of (X1 union X2) st

( H is closed & x in H & H c= F1 \/ F2 )

let X1, X2 be non empty SubSpace of X; :: thesis: for x being Point of (X1 union X2)

for F1 being Subset of X1

for F2 being Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 holds

ex H being Subset of (X1 union X2) st

( H is closed & x in H & H c= F1 \/ F2 )

let x be Point of (X1 union X2); :: thesis: for F1 being Subset of X1

for F2 being Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 holds

ex H being Subset of (X1 union X2) st

( H is closed & x in H & H c= F1 \/ F2 )

let F1 be Subset of X1; :: thesis: for F2 being Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 holds

ex H being Subset of (X1 union X2) st

( H is closed & x in H & H c= F1 \/ F2 )

let F2 be Subset of X2; :: thesis: ( F1 is closed & x in F1 & F2 is closed & x in F2 implies ex H being Subset of (X1 union X2) st

( H is closed & x in H & H c= F1 \/ F2 ) )

assume that

A1: F1 is closed and

A2: x in F1 and

A3: F2 is closed and

A4: x in F2 ; :: thesis: ex H being Subset of (X1 union X2) st

( H is closed & x in H & H c= F1 \/ F2 )

A5: X1 is SubSpace of X1 union X2 by TSEP_1:22;

then reconsider C1 = the carrier of X1 as Subset of (X1 union X2) by TSEP_1:1;

consider H1 being Subset of (X1 union X2) such that

A6: H1 is closed and

A7: H1 /\ ([#] X1) = F1 by A1, A5, PRE_TOPC:13;

A8: x in H1 by A2, A7, XBOOLE_0:def 4;

A9: X2 is SubSpace of X1 union X2 by TSEP_1:22;

then reconsider C2 = the carrier of X2 as Subset of (X1 union X2) by TSEP_1:1;

consider H2 being Subset of (X1 union X2) such that

A10: H2 is closed and

A11: H2 /\ ([#] X2) = F2 by A3, A9, PRE_TOPC:13;

A12: x in H2 by A4, A11, XBOOLE_0:def 4;

take H = H1 /\ H2; :: thesis: ( H is closed & x in H & H c= F1 \/ F2 )

A13: ( H /\ C1 c= H1 /\ C1 & H /\ C2 c= H2 /\ C2 ) by XBOOLE_1:17, XBOOLE_1:26;

the carrier of (X1 union X2) = C1 \/ C2 by TSEP_1:def 2;

then H = H /\ (C1 \/ C2) by XBOOLE_1:28

.= (H /\ C1) \/ (H /\ C2) by XBOOLE_1:23 ;

hence ( H is closed & x in H & H c= F1 \/ F2 ) by A6, A7, A10, A11, A13, A8, A12, XBOOLE_0:def 4, XBOOLE_1:13; :: thesis: verum

for x being Point of (X1 union X2)

for F1 being Subset of X1

for F2 being Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 holds

ex H being Subset of (X1 union X2) st

( H is closed & x in H & H c= F1 \/ F2 )

let X1, X2 be non empty SubSpace of X; :: thesis: for x being Point of (X1 union X2)

for F1 being Subset of X1

for F2 being Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 holds

ex H being Subset of (X1 union X2) st

( H is closed & x in H & H c= F1 \/ F2 )

let x be Point of (X1 union X2); :: thesis: for F1 being Subset of X1

for F2 being Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 holds

ex H being Subset of (X1 union X2) st

( H is closed & x in H & H c= F1 \/ F2 )

let F1 be Subset of X1; :: thesis: for F2 being Subset of X2 st F1 is closed & x in F1 & F2 is closed & x in F2 holds

ex H being Subset of (X1 union X2) st

( H is closed & x in H & H c= F1 \/ F2 )

let F2 be Subset of X2; :: thesis: ( F1 is closed & x in F1 & F2 is closed & x in F2 implies ex H being Subset of (X1 union X2) st

( H is closed & x in H & H c= F1 \/ F2 ) )

assume that

A1: F1 is closed and

A2: x in F1 and

A3: F2 is closed and

A4: x in F2 ; :: thesis: ex H being Subset of (X1 union X2) st

( H is closed & x in H & H c= F1 \/ F2 )

A5: X1 is SubSpace of X1 union X2 by TSEP_1:22;

then reconsider C1 = the carrier of X1 as Subset of (X1 union X2) by TSEP_1:1;

consider H1 being Subset of (X1 union X2) such that

A6: H1 is closed and

A7: H1 /\ ([#] X1) = F1 by A1, A5, PRE_TOPC:13;

A8: x in H1 by A2, A7, XBOOLE_0:def 4;

A9: X2 is SubSpace of X1 union X2 by TSEP_1:22;

then reconsider C2 = the carrier of X2 as Subset of (X1 union X2) by TSEP_1:1;

consider H2 being Subset of (X1 union X2) such that

A10: H2 is closed and

A11: H2 /\ ([#] X2) = F2 by A3, A9, PRE_TOPC:13;

A12: x in H2 by A4, A11, XBOOLE_0:def 4;

take H = H1 /\ H2; :: thesis: ( H is closed & x in H & H c= F1 \/ F2 )

A13: ( H /\ C1 c= H1 /\ C1 & H /\ C2 c= H2 /\ C2 ) by XBOOLE_1:17, XBOOLE_1:26;

the carrier of (X1 union X2) = C1 \/ C2 by TSEP_1:def 2;

then H = H /\ (C1 \/ C2) by XBOOLE_1:28

.= (H /\ C1) \/ (H /\ C2) by XBOOLE_1:23 ;

hence ( H is closed & x in H & H c= F1 \/ F2 ) by A6, A7, A10, A11, A13, A8, A12, XBOOLE_0:def 4, XBOOLE_1:13; :: thesis: verum