reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;
reserve u for UnOp of G;
reserve A for Abelian addGroup;
reserve a,b for Element of A;
reserve x for object;
reserve y,y1,y2,Y,Z for set;
reserve k for Nat;
reserve G for addGroup;
reserve a,g,h for Element of G;
reserve A for Subset of G;
reserve G for non empty addMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for addGroup-like non empty addMagma;
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 addGroup;
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 addF 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 addF of H1 = h||A & the addF of H2 = h||B by DefA5;
  hence thesis by A2,FUNCT_1:51;
end;
