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 Th73:
  for G,H being strict Group holds G,H are_isomorphic implies card G = card H
proof
  let G,H be strict Group;
  assume
A1: G,H are_isomorphic;
  then consider h being Homomorphism of G,H such that
A2: h is bijective;
  consider g1 being Homomorphism of H,G such that
A3: g1 is bijective by A1,Def11;
  Image g1 = G by A3,Th57;
  then
A4: card G c= card H by Th52;
  Image h = H by A2,Th57;
  hence thesis by A4,Th52;
end;
