theorem Th57:
  the carrier of H1 c= the carrier of H2 implies H1 is Subgroup of H2
proof
  set A = the carrier of H1;
  set B = the carrier of H2;
  set h = the multF of G;
  assume
A1: the carrier of H1 c= the carrier of H2;
  hence the carrier of H1 c= the carrier of H2;
A2: [:A,A:] c= [:B,B:] by A1,ZFMISC_1:96;
  the multF of H1 = h||A & the multF of H2 = h||B by Def5;
  hence thesis by A2,FUNCT_1:51;
end;
