let T1, T2 be TopSpace; :: thesis: for A1 being Subset of T1
for A2 being Subset of T2 st A1,A2 are_homeomorphic holds
( A1 is finite-ind iff A2 is finite-ind )
let A1 be Subset of T1; :: thesis: for A2 being Subset of T2 st A1,A2 are_homeomorphic holds
( A1 is finite-ind iff A2 is finite-ind )
let A2 be Subset of T2; :: thesis: ( A1,A2 are_homeomorphic implies ( A1 is finite-ind iff A2 is finite-ind ) )
assume A1,A2 are_homeomorphic ; :: thesis: ( A1 is finite-ind iff A2 is finite-ind )
then A1: T1 | A1,T2 | A2 are_homeomorphic by METRIZTS:def 1;

hereby :: thesis: ( A2 is finite-ind implies A1 is finite-ind )

assume A1 is finite-ind ; :: thesis: A2 is finite-ind
then T2 | A2 is finite-ind by A1, Lm9;
hence A2 is finite-ind by Th18; :: thesis: verum

end;

assume A2 is finite-ind ; :: thesis: A1 is finite-ind
then T1 | A1 is finite-ind by A1, Lm9;
hence A1 is finite-ind by Th18; :: thesis: verum