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 & x in F1 ) and
A2: ( F2 is closed & x in F2 ) ; :: thesis: ex H being Subset of (X1 union X2) st
( H is closed & x in H & H c= F1 \/ F2 )
A3: X1 is SubSpace of X1 union X2 by TSEP_1:22;
then consider H1 being Subset of (X1 union X2) such that
A4: ( H1 is closed & H1 /\ ([#] X1) = F1 ) by A1, PRE_TOPC:43;
A5: X2 is SubSpace of X1 union X2 by TSEP_1:22;
then consider H2 being Subset of (X1 union X2) such that
A6: ( H2 is closed & H2 /\ ([#] X2) = F2 ) by A2, PRE_TOPC:43;
take H = H1 /\ H2; :: thesis: ( H is closed & x in H & H c= F1 \/ F2 )
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: H = H /\ (C1 \/ C2) by XBOOLE_1:28
.= (H /\ C1) \/ (H /\ C2) by XBOOLE_1:23 ;
A8: ( H /\ C1 c= H1 /\ C1 & H /\ C2 c= H2 /\ C2 ) by XBOOLE_1:17, XBOOLE_1:26;
( x in H1 & x in H2 ) by A1, A2, A4, A6, XBOOLE_0:def 4;
hence ( H is closed & x in H & H c= F1 \/ F2 ) by A4, A6, A7, A8, TOPS_1:35, XBOOLE_0:def 4, XBOOLE_1:13; :: thesis: verum