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
  add_inverse(H) = add_inverse(G) | the carrier of H
proof
A1: (the carrier of G) /\ (the carrier of H) = the carrier of H
    by DefA5,XBOOLE_1:28;
A2: now
    let x be object;
    assume x in dom(add_inverse(H));
    then reconsider a = x as Element of H;
    reconsider b = a as Element of G by Th41,STRUCT_0:def 5;
    thus add_inverse(H).x = -a by Def6
      .= -b by Th48
      .= add_inverse(G).x by Def6;
  end;
  dom(add_inverse(H)) = the carrier of H & dom(add_inverse(G)) = the carrier
  of G by FUNCT_2:def 1;
  hence thesis by A1,A2,FUNCT_1:46;
end;
