theorem
  G1 is finite-ind & G2 c= G1 implies G2 is finite-ind & ind G2 <= ind G1
proof
  assume that
A1: G1 is finite-ind and
A2: G2 c=G1;
A3: -1<=ind G1 by A1,Th11;
  then for A st A in G2 holds A is finite-ind & ind A<=ind G1 by A1,A2,Th11;
  hence thesis by A3,Th11;
end;
