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;

theorem
  for G being Abelian addGroup, A,B being Subset of G holds
  -(A + B) = -A + -B
proof
  let G be Abelian addGroup, A,B be Subset of G;
  thus -(A + B) c= -A + -B
  proof
    let x be object;
    assume x in -(A + B);
    then consider g being Element of G such that
A1: x = -g and
A2: g in A + B;
    consider g1,g2 being Element of G such that
A3: g = g1 + g2 and
A4: g1 in A & g2 in B by A2;
A5: -g1 in -A & -g2 in -B by A4;
    x = -g1 + -g2 by A1,A3,Th44;
    hence thesis by A5;
  end;
  let x be object;
  assume x in -A + -B;
  then consider g1,g2 being Element of G such that
A6: x = g1 + g2 and
A7: g1 in -A and
A8: g2 in -B;
  consider b being Element of G such that
A9: g2 = -b and
A10: b in B by A8;
  consider a being Element of G such that
A11: g1 = -a and
A12: a in A by A7;
A13: a + b in A + B by A12,A10;
  x = -(a + b) by A6,A11,A9,Th44;
  hence thesis by A13;
end;
