let GX be TopSpace; :: thesis: for X9 being SubSpace of GX

for P, Q being Subset of GX

for P1, Q1 being Subset of X9 st P = P1 & Q = Q1 & P \/ Q c= [#] X9 & P,Q are_separated holds

P1,Q1 are_separated

let X9 be SubSpace of GX; :: thesis: for P, Q being Subset of GX

for P1, Q1 being Subset of X9 st P = P1 & Q = Q1 & P \/ Q c= [#] X9 & P,Q are_separated holds

P1,Q1 are_separated

let P, Q be Subset of GX; :: thesis: for P1, Q1 being Subset of X9 st P = P1 & Q = Q1 & P \/ Q c= [#] X9 & P,Q are_separated holds

P1,Q1 are_separated

let P1, Q1 be Subset of X9; :: thesis: ( P = P1 & Q = Q1 & P \/ Q c= [#] X9 & P,Q are_separated implies P1,Q1 are_separated )

assume that

A1: P = P1 and

A2: Q = Q1 and

A3: P \/ Q c= [#] X9 ; :: thesis: ( not P,Q are_separated or P1,Q1 are_separated )

A4: Q c= P \/ Q by XBOOLE_1:7;

P c= P \/ Q by XBOOLE_1:7;

then reconsider P2 = P, Q2 = Q as Subset of X9 by A3, A4, XBOOLE_1:1;

assume that

A5: (Cl P) /\ Q = {} and

A6: P /\ (Cl Q) = {} ; :: according to XBOOLE_0:def 7,CONNSP_1:def 1 :: thesis: P1,Q1 are_separated

P2 /\ (Cl Q2) = P2 /\ (([#] X9) /\ (Cl Q)) by PRE_TOPC:17

.= (P2 /\ ([#] X9)) /\ (Cl Q) by XBOOLE_1:16

.= P /\ (Cl Q) by XBOOLE_1:28 ;

then A7: P2 misses Cl Q2 by A6;

Cl P2 = (Cl P) /\ ([#] X9) by PRE_TOPC:17;

then (Cl P2) /\ Q2 = (Cl P) /\ (Q2 /\ ([#] X9)) by XBOOLE_1:16

.= (Cl P) /\ Q by XBOOLE_1:28 ;

then Cl P2 misses Q2 by A5;

hence P1,Q1 are_separated by A1, A2, A7; :: thesis: verum

for P, Q being Subset of GX

for P1, Q1 being Subset of X9 st P = P1 & Q = Q1 & P \/ Q c= [#] X9 & P,Q are_separated holds

P1,Q1 are_separated

let X9 be SubSpace of GX; :: thesis: for P, Q being Subset of GX

for P1, Q1 being Subset of X9 st P = P1 & Q = Q1 & P \/ Q c= [#] X9 & P,Q are_separated holds

P1,Q1 are_separated

let P, Q be Subset of GX; :: thesis: for P1, Q1 being Subset of X9 st P = P1 & Q = Q1 & P \/ Q c= [#] X9 & P,Q are_separated holds

P1,Q1 are_separated

let P1, Q1 be Subset of X9; :: thesis: ( P = P1 & Q = Q1 & P \/ Q c= [#] X9 & P,Q are_separated implies P1,Q1 are_separated )

assume that

A1: P = P1 and

A2: Q = Q1 and

A3: P \/ Q c= [#] X9 ; :: thesis: ( not P,Q are_separated or P1,Q1 are_separated )

A4: Q c= P \/ Q by XBOOLE_1:7;

P c= P \/ Q by XBOOLE_1:7;

then reconsider P2 = P, Q2 = Q as Subset of X9 by A3, A4, XBOOLE_1:1;

assume that

A5: (Cl P) /\ Q = {} and

A6: P /\ (Cl Q) = {} ; :: according to XBOOLE_0:def 7,CONNSP_1:def 1 :: thesis: P1,Q1 are_separated

P2 /\ (Cl Q2) = P2 /\ (([#] X9) /\ (Cl Q)) by PRE_TOPC:17

.= (P2 /\ ([#] X9)) /\ (Cl Q) by XBOOLE_1:16

.= P /\ (Cl Q) by XBOOLE_1:28 ;

then A7: P2 misses Cl Q2 by A6;

Cl P2 = (Cl P) /\ ([#] X9) by PRE_TOPC:17;

then (Cl P2) /\ Q2 = (Cl P) /\ (Q2 /\ ([#] X9)) by XBOOLE_1:16

.= (Cl P) /\ Q by XBOOLE_1:28 ;

then Cl P2 misses Q2 by A5;

hence P1,Q1 are_separated by A1, A2, A7; :: thesis: verum