theorem Th21:
  F = B & B is finite-ind implies F is finite-ind & ind F = ind B
proof
  assume that
A1: F=B and
A2: B is finite-ind;
A3: T|A|F=T|B by A1,METRIZTS:9;
  then F is finite-ind by A2,Th18;
  then ind F=ind(T|A|F) by Lm5
    .=ind(T|B) by A1,METRIZTS:9
    .=ind B by A2,Lm5;
  hence thesis by A2,A3,Th18;
end;
