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
  (-A) * B = -(A * B)
proof
  thus (-A) * B c= -(A * B)
  proof
    let x be object;
    assume x in (-A) * B;
    then consider a,b such that
A1: x = a * b and
A2: a in (-A) and
A3: b in B;
    consider c such that
A4: a = -c & c in A by A2;
    x = -(c * b) & c * b in A * B by A1,A3,A4,Th26;
    hence thesis;
  end;
  let x be object;
  assume x in -(A * B);
  then consider a such that
A5: x = (-a) and
A6: a in A * B;
  consider b,c such that
A7: a = b * c and
A8: b in A and
A9: c in B by A6;
A10: (-b) in (-A) by A8;
  x = (-b) * c by A5,A7,Th26;
  hence thesis by A9,A10;
end;
