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

theorem Th15:
  for X being set, u,v being Subset of X, x being Element of X
  holds (u \+\ v)@x = u@x + v@x
proof
  let X be set, u,v be Subset of X, x be Element of X;
  per cases;
  suppose
A1: x in u \+\ v;
    then
A2: (u \+\ v)@x = 1.Z_2 by Def3;
    per cases;
    suppose
A3:   x in u;
      then not x in v by A1,XBOOLE_0:1;
      then
A4:   v@x = 0.Z_2 by Def3;
      u@x = 1.Z_2 by A3,Def3;
      hence thesis by A2,A4,RLVECT_1:4;
    end;
    suppose
A5:   not x in u;
      then x in v by A1,XBOOLE_0:1;
      then
A6:   v@x = 1.Z_2 by Def3;
      u@x = 0.Z_2 by A5,Def3;
      hence thesis by A2,A6,RLVECT_1:4;
    end;
  end;
  suppose
A7: not x in u \+\ v;
    then
A8: (u \+\ v)@x = 0.Z_2 by Def3;
    per cases;
    suppose
      x in u;
      then x in v & u@x = 1.Z_2 by A7,Def3,XBOOLE_0:1;
      hence thesis by A8,Def3,Th7;
    end;
    suppose
A9:   not x in u;
      then not x in v by A7,XBOOLE_0:1;
      then
A10:  v@x = 0.Z_2 by Def3;
      u@x = 0.Z_2 by A9,Def3;
      hence thesis by A8,A10,RLVECT_1:4;
    end;
  end;
end;
