let T1, T2 be strict TopSpace; :: thesis: ( the carrier of T1 = X & ( for A being Subset of T1 holds Int A = IFEQ (A,X,A,(A /\ X0)) ) & the carrier of T2 = X & ( for A being Subset of T2 holds Int A = IFEQ (A,X,A,(A /\ X0)) ) implies T1 = T2 )

assume that

A14: the carrier of T1 = X and

A15: for A being Subset of T1 holds Int A = IFEQ (A,X,A,(A /\ X0)) and

A16: the carrier of T2 = X and

A17: for A being Subset of T2 holds Int A = IFEQ (A,X,A,(A /\ X0)) ; :: thesis: T1 = T2

assume that

A14: the carrier of T1 = X and

A15: for A being Subset of T1 holds Int A = IFEQ (A,X,A,(A /\ X0)) and

A16: the carrier of T2 = X and

A17: for A being Subset of T2 holds Int A = IFEQ (A,X,A,(A /\ X0)) ; :: thesis: T1 = T2

now :: thesis: for A1 being Subset of T1

for A2 being Subset of T2 st A1 = A2 holds

Int A1 = Int A2

hence
T1 = T2
by A14, A16, Th10; :: thesis: verumfor A2 being Subset of T2 st A1 = A2 holds

Int A1 = Int A2

let A1 be Subset of T1; :: thesis: for A2 being Subset of T2 st A1 = A2 holds

Int A1 = Int A2

let A2 be Subset of T2; :: thesis: ( A1 = A2 implies Int A1 = Int A2 )

assume A1 = A2 ; :: thesis: Int A1 = Int A2

hence Int A1 = IFEQ (A2,X,A2,(A2 /\ X0)) by A15

.= Int A2 by A17 ;

:: thesis: verum

end;Int A1 = Int A2

let A2 be Subset of T2; :: thesis: ( A1 = A2 implies Int A1 = Int A2 )

assume A1 = A2 ; :: thesis: Int A1 = Int A2

hence Int A1 = IFEQ (A2,X,A2,(A2 /\ X0)) by A15

.= Int A2 by A17 ;

:: thesis: verum