reserve S for 1-sorted,
  i for Element of NAT,
  p for FinSequence,
  X for set;

theorem Th17:
  for a being Element of Z_2, c,d being Subset of X holds a \*\ (c
  \+\ d) = (a \*\ c) \+\ (a \*\ d)
proof
  let a be Element of Z_2, c,d be Subset of X;
  per cases by Th5,Th6,CARD_1:50,TARSKI:def 2;
  suppose
A1: a = 0.Z_2;
    then a \*\ (c \+\ d) = {}X & a \*\ c = {}X by Def4;
    hence thesis by A1,Def4;
  end;
  suppose
A2: a = 1.Z_2;
    then a \*\ (c \+\ d) = c \+\ d & a \*\ c = c by Def4;
    hence thesis by A2,Def4;
  end;
end;
