 reserve x for object;
 reserve G for non empty 1-sorted;
 reserve A for Subset of G;
 reserve y,y1,y2,Y,Z for set;
 reserve k for Nat;
 reserve G for Group;
 reserve a,g,h for Element of G;
 reserve A for Subset of G;
reserve G for non empty multMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;

theorem Th10:
  G is associative implies A * B * C = A * (B * C)
proof
  assume
A1: G is associative;
  thus A * B * C c= A * (B * C)
  proof
    let x be object;
    assume x in A * B * C;
    then consider g,h such that
A2: x = g * h and
A3: g in A * B and
A4: h in C;
    consider g1,g2 such that
A5: g = g1 * g2 and
A6: g1 in A and
A7: g2 in B by A3;
    x = g1 * (g2 * h) & g2 * h in B * C by A1,A2,A4,A5,A7;
    hence thesis by A6;
  end;
  let x be object;
  assume x in A * (B * C);
  then consider g,h such that
A8: x = g * h and
A9: g in A and
A10: h in B * C;
  consider g1,g2 such that
A11: h = g1 * g2 and
A12: g1 in B and
A13: g2 in C by A10;
A14: g * g1 in A * B by A9,A12;
  x = g * g1 * g2 by A1,A8,A11;
  hence thesis by A13,A14;
end;
