reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;
reserve N for normal Subgroup of G;
reserve S,T1,T2 for Element of G./.N;
reserve g,h for Homomorphism of G,H;
reserve h1 for Homomorphism of H,I;

theorem Th74:
  G,H are_isomorphic & G is finite implies H is finite
proof
  assume that
A1: G,H are_isomorphic and
A2: G is finite;
  consider h such that
A3: h is bijective by A1;
  rng h = the carrier of H by A3,FUNCT_2:def 3;
  hence thesis by A2;
end;
