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;
reserve x,y,y1,y2 for set;
reserve G for addGroup;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem Th1:
  a + b + (-b) = a & a + (-b) + b = a & (-b) + b + a = a & b + (-b) + a = a
  & a + (b + (-b)) = a & a + ((-b) + b) = a & (-b) + (b + a) = a &
  b + ((-b) + a) = a
proof
  thus
A1: a + b + (-b) = a + (b + (-b)) by RLVECT_1:def 3
    .= a + 0_G by Def5
    .= a by Def4;
  thus
A2: a + (-b) + b = a + ((-b) + b) by RLVECT_1:def 3
    .= a + 0_G by Def5
    .= a by Def4;
  thus
A3: (-b) + b + a = 0_G + a by Def5
    .= a by Def4;
  thus b + (-b) + a = 0_G + a by Def5
    .= a by Def4;
  hence thesis by A1,A2,A3,RLVECT_1:def 3;
end;
