let T1, T2 be strict TopSpace; :: thesis: ( the carrier of T1 = X & ( for A being Subset of T1 holds Cl A = IFEQ A,{} ,A,(A \/ ({x0} /\ X)) ) & the carrier of T2 = X & ( for A being Subset of T2 holds Cl A = IFEQ A,{} ,A,(A \/ ({x0} /\ X)) ) implies T1 = T2 )
assume that
A8: the carrier of T1 = X and
A9: for A being Subset of T1 holds Cl A = IFEQ A,{} ,A,(A \/ ({x0} /\ X)) and
A10: the carrier of T2 = X and
A11: for A being Subset of T2 holds Cl A = IFEQ A,{} ,A,(A \/ ({x0} /\ X)) ; :: thesis: T1 = T2

now

let A1 be Subset of T1; :: thesis: for A2 being Subset of T2 st A1 = A2 holds
Cl A1 = Cl A2
let A2 be Subset of T2; :: thesis: ( A1 = A2 implies Cl A1 = Cl A2 )
assume A1 = A2 ; :: thesis: Cl A1 = Cl A2
hence Cl A1 = IFEQ A2,{} ,A2,(A2 \/ ({x0} /\ X)) by A9
.= Cl A2 by A11 ;
:: thesis: verum

end;

hence T1 = T2 by A8, A10, Th8; :: thesis: verum