let FT be non empty RelStr ; :: thesis: for X' being non empty SubSpace of FT
for P1, Q1 being Subset of FT
for P, Q being Subset of X' st P = P1 & Q = Q1 & P,Q are_separated holds
P1,Q1 are_separated
let X' be non empty SubSpace of FT; :: thesis: for P1, Q1 being Subset of FT
for P, Q being Subset of X' st P = P1 & Q = Q1 & P,Q are_separated holds
P1,Q1 are_separated
let P1, Q1 be Subset of FT; :: thesis: for P, Q being Subset of X' st P = P1 & Q = Q1 & P,Q are_separated holds
P1,Q1 are_separated
let P, Q be Subset of X'; :: thesis: ( P = P1 & Q = Q1 & P,Q are_separated implies P1,Q1 are_separated )
assume that
A1: P = P1 and
A2: Q = Q1 ; :: thesis: ( not P,Q are_separated or P1,Q1 are_separated )
assume P,Q are_separated ; :: thesis: P1,Q1 are_separated
then ( P ^b misses Q & P misses Q ^b ) by FINTOPO4:def 1;
then A3: ( (P ^b ) /\ Q = {} & P /\ (Q ^b ) = {} ) by XBOOLE_0:def 7;
reconsider P2 = P, Q2 = Q as Subset of FT by Th10;
A4: (P ^b ) /\ Q = ((P2 ^b ) /\ ([#] X')) /\ Q by Th13
.= (P2 ^b ) /\ (Q /\ ([#] X')) by XBOOLE_1:16
.= (P2 ^b ) /\ Q2 by XBOOLE_1:28 ;
P /\ (Q ^b ) = P /\ (([#] X') /\ (Q2 ^b )) by Th13
.= (P /\ ([#] X')) /\ (Q2 ^b ) by XBOOLE_1:16
.= P2 /\ (Q2 ^b ) by XBOOLE_1:28 ;
then ( P2 ^b misses Q2 & P2 misses Q2 ^b ) by A3, A4, XBOOLE_0:def 7;
hence P1,Q1 are_separated by A1, A2, FINTOPO4:def 1; :: thesis: verum