 reserve x for object;
 reserve G for non empty 1-sorted;
 reserve A for Subset of G;
 reserve y,y1,y2,Y,Z for set;
 reserve k for Nat;
 reserve G for Group;
 reserve a,g,h for Element of G;
 reserve A for Subset of G;
reserve G for non empty multMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for Group-like non empty multMagma;
reserve h,g,g1,g2 for Element of G;
reserve A for Subset of G;
reserve H for Subgroup of G;
reserve h,h1,h2 for Element of H;
reserve G,G1,G2,G3 for Group;
reserve a,a1,a2,b,b1,b2,g,g1,g2 for Element of G;
reserve A,B for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve h,h1,h2 for Element of H;

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;
