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

theorem Th18:
  for a,b being Element of Z_2, c being Subset of X holds (a+b)
  \*\ c = (a \*\ c) \+\ (b \*\ c)
proof
  let a,b be Element of Z_2, c be Subset of X;
  per cases by Th5,Th6,CARD_1:50,TARSKI:def 2;
  suppose
A1: a = 0.Z_2;
    then a \*\ c = {}X by Def4;
    hence thesis by A1,RLVECT_1:4;
  end;
  suppose
A2: a = 1.Z_2;
    per cases by Th5,Th6,CARD_1:50,TARSKI:def 2;
    suppose
A3:   b = 0.Z_2;
      then b \*\ c = {}X by Def4;
      hence thesis by A3,RLVECT_1:4;
    end;
    suppose
A4:   b = 1.Z_2;
A5:   c \+\ c = {}X by XBOOLE_1:92;
      b \*\ c = c by A4,Def4;
      hence thesis by A2,A4,A5,Def4,Th7;
    end;
  end;
end;
