reserve x,y,y1,y2 for set;
reserve G for Group;
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 GROUP_1:def 3
    .= a * 1_G by GROUP_1:def 5
    .= a by GROUP_1:def 4;
  thus
A2: a * b" * b = a * (b" * b) by GROUP_1:def 3
    .= a * 1_G by GROUP_1:def 5
    .= a by GROUP_1:def 4;
  thus
A3: b" * b * a = 1_G * a by GROUP_1:def 5
    .= a by GROUP_1:def 4;
  thus b * b" * a = 1_G * a by GROUP_1:def 5
    .= a by GROUP_1:def 4;
  hence thesis by A1,A2,A3,GROUP_1:def 3;
end;
