reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;
reserve A,B for set;

theorem
  for f,g being Function, B,C being set st dom f c= B & dom g c= C & B
  misses C holds (f +* g)|B = f & (f +* g)|C = g
proof
  let f,g be Function, B,C be set;
  assume that
A1: dom f c= B and
A2: dom g c= C and
A3: B misses C;
  dom f misses C by A1,A3,XBOOLE_1:63;
  then dom f /\ C = {};
  then dom (f|C) = {} by RELAT_1:61;
  then
A4: f|C = {};
  dom g misses B by A2,A3,XBOOLE_1:63;
  then dom g /\ B = {};
  then dom (g|B) = {} by RELAT_1:61;
  then
A5: g|B = {};
  thus (f +* g)|B = f|B +* g|B by Th71
    .= f|B by A5
    .= f by A1,RELAT_1:68;
  thus (f +* g)|C = f|C +* g|C by Th71
    .= g|C by A4
    .= g by A2,RELAT_1:68;
end;
