reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;
reserve F,G for Cardinal-Function;
reserve A,B for set;
reserve A,B for Ordinal;
reserve n,k for Nat;

theorem
  for f,g being non-empty Function
  for x being Element of product (f+*g) holds x|dom g in product g
proof
  let f,g be non-empty Function;
  let x be Element of product (f+*g);
A1: dom x = dom (f+*g) by Th9;
A2: dom (f+*g) = dom f \/ dom g by FUNCT_4:def 1;
  then
A3: dom g c= dom x by A1,XBOOLE_1:7;
A4: dom (x|dom g) = dom g by A1,A2,RELAT_1:62,XBOOLE_1:7;
  now
    let z be object;
    assume
A5: z in dom (x|dom g);
    then
A6: (x|dom g).z = x.z by FUNCT_1:47;
    (f+*g).z = g.z by A4,A5,FUNCT_4:13;
    hence (x|dom g).z in g.z by A1,A3,A4,A5,A6,Th9;
  end;
  hence thesis by A4,Th9;
end;
